{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T01:41:18Z","timestamp":1776044478063,"version":"3.50.1"},"reference-count":90,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T00:00:00Z","timestamp":1776038400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T00:00:00Z","timestamp":1776038400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Evol. Intel."],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s12065-026-01187-2","type":"journal-article","created":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T00:44:18Z","timestamp":1776041058000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reduced order modeling of commensurate fractional order systems using Arnoldi\u2013Krylov subspace optimization technique"],"prefix":"10.1007","volume":"19","author":[{"given":"Anirban","family":"Bose","sequence":"first","affiliation":[]},{"given":"Arindam","family":"Mondal","sequence":"additional","affiliation":[]},{"given":"Chandan","family":"Das","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,4,13]]},"reference":[{"issue":"6","key":"1187_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.25103\/jestr.126.01","volume":"12","author":"K Kothari","year":"2019","unstructured":"Kothari K, Mehta UV, Prasad R (2019) Fractional-order system modeling and its applications. Journal of Engineering Science and Technology Review 12(6):1\u201310","journal-title":"Journal of Engineering Science and Technology Review"},{"key":"1187_CR2","doi-asserted-by":"crossref","unstructured":"Caponetto R, Dongola G, Fortuna L, Petras I (2010) Fractional Order Systems: Modeling and Control Applications, vol 72. World scientific, Singapore","DOI":"10.1142\/7709"},{"key":"1187_CR3","unstructured":"Podlubny I (1994) The Laplace Transform Method for Linear Differential Equations of the Fractional Order. Slovak Academy of Sciences, Kosice, Slovakia . 10.48550\/arXiv.funct-an\/9710005"},{"issue":"3","key":"1187_CR4","first-page":"231","volume":"3","author":"BM Vinagre","year":"2000","unstructured":"Vinagre BM, Podlubny I, Feliu V (2000) Some approximations of fractional order operators used in control theory and applications. J Fract Calc Appl Anal 3(3):231\u2013248","journal-title":"J Fract Calc Appl Anal"},{"key":"1187_CR5","doi-asserted-by":"publisher","first-page":"1941011","DOI":"10.1142\/S179396231941011X","volume":"10","author":"BB Alagoz","year":"2019","unstructured":"Alagoz BB, Tepljakov A, Ates A, Petlenkov E, Yeroglu C (2019) Time-domain identification of one noninteger order plus time delay models from step response measurements. Int J Model Simul Sci Comput 10:1941011. https:\/\/doi.org\/10.1142\/S179396231941011X","journal-title":"Int J Model Simul Sci Comput"},{"key":"1187_CR6","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1016\/j.enbuild.2016.09.030","volume":"133","author":"L Chen","year":"2016","unstructured":"Chen L, Basu B, McCabe D (2016) Fractional order models for system identification of thermal dynamics of buildings. Energy Build 133:381\u2013388. https:\/\/doi.org\/10.1016\/j.enbuild.2016.09.030","journal-title":"Energy Build"},{"key":"1187_CR7","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1002\/cey2.360","volume":"6","author":"W Li","year":"2024","unstructured":"Li W, Huang QA, Bai Y, Wang J, Wang L, Liu Y, Zhang J (2024) Model reduction of fractional impedance spectra for time-frequency analysis of batteries, fuel cells, and supercapacitors. Carbon Energy 6:360. https:\/\/doi.org\/10.1002\/cey2.360","journal-title":"Carbon Energy"},{"key":"1187_CR8","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1016\/j.aej.2023.08.007","volume":"95","author":"KS Nisar","year":"2024","unstructured":"Nisar KS, Farman M, Abdel-Aty M, Ravichandran C (2024) A review of fractional order epidemic models for life sciences problems: Past, present and future. Alex Eng J 95:283\u2013305. https:\/\/doi.org\/10.1016\/j.aej.2023.08.007","journal-title":"Alex Eng J"},{"key":"1187_CR9","doi-asserted-by":"publisher","first-page":"403","DOI":"10.3390\/fractalfract8040403","volume":"8","author":"AP Singh","year":"2024","unstructured":"Singh AP, Bingi K (2024) Applications of fractional-order calculus in robotics. Fractal Fract 8:403. https:\/\/doi.org\/10.3390\/fractalfract8040403","journal-title":"Fractal Fract"},{"key":"1187_CR10","doi-asserted-by":"publisher","unstructured":"Zhang JR, Lu JG, Zhu Z (2024) Robust h$$\\infty $$ model reduction for fractional-order two-dimensional continuous roesser system: The 1 $$<\\epsilon <2$$ case. In: Proceedings of the 2024 36th Chinese Control and Decision Conference (CCDC), Xi\u2019an, China, pp. 1100\u20131105 . https:\/\/doi.org\/10.1109\/CCDC.2024.1234567","DOI":"10.1109\/CCDC.2024.1234567"},{"key":"1187_CR11","doi-asserted-by":"publisher","first-page":"1850176","DOI":"10.1142\/S0218126618501762","volume":"27","author":"A Tepljakov","year":"2018","unstructured":"Tepljakov A, Alagoz BB, Gonzalez E, Petlenkov E, Yeroglu C (2018) Model reference adaptive control scheme for retuning method-based fractional-order pid control with disturbance rejection applied to closed-loop control of a magnetic levitation system. J Circ Syst Comput 27:1850176. https:\/\/doi.org\/10.1142\/S0218126618501762","journal-title":"J Circ Syst Comput"},{"key":"1187_CR12","doi-asserted-by":"publisher","first-page":"3440","DOI":"10.1007\/s00034-024-02552-3","volume":"43","author":"S Gehlaut","year":"2024","unstructured":"Gehlaut S, Kumar D (2024) Salp swarm optimization-based approximation of fractional-order systems with guaranteed stability. Circ Syst Signal Process 43:3440\u20133460. https:\/\/doi.org\/10.1007\/s00034-024-02552-3","journal-title":"Circ Syst Signal Process"},{"key":"1187_CR13","doi-asserted-by":"publisher","first-page":"4236","DOI":"10.1109\/TSMC.2024.3314873","volume":"54","author":"C Yang","year":"2024","unstructured":"Yang C, Fan Z, Xia Y (2024) Convex model-based reduced-order model for uncertain control systems. IEEE Trans Syst Man Cybern Syst 54:4236\u20134246. https:\/\/doi.org\/10.1109\/TSMC.2024.3314873","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"1187_CR14","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.amc.2017.10.038","volume":"322","author":"RJ Liu","year":"2018","unstructured":"Liu RJ, Nie ZY, Wu M, She J (2018) Robust disturbance rejection for uncertain fractional-order systems. Appl Math Comput 322:79\u201388. https:\/\/doi.org\/10.1016\/j.amc.2017.10.038","journal-title":"Appl Math Comput"},{"key":"1187_CR15","doi-asserted-by":"publisher","first-page":"460","DOI":"10.3390\/fractalfract7070460","volume":"7","author":"A Y\u00fcce","year":"2023","unstructured":"Y\u00fcce A (2023) An approximation method for fractional-order models using quadratic systems and equilibrium optimizer. Fractal Fract 7:460. https:\/\/doi.org\/10.3390\/fractalfract7070460","journal-title":"Fractal Fract"},{"key":"1187_CR16","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.arcontrol.2020.03.005","volume":"49","author":"FN Deniz","year":"2020","unstructured":"Deniz FN, Alagoz BB, Tan N, Koseoglu M (2020) Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses. Annu Rev Control 49:239\u2013257. https:\/\/doi.org\/10.1016\/j.arcontrol.2020.03.005","journal-title":"Annu Rev Control"},{"key":"1187_CR17","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.ifacol.2018.06.098","volume":"51","author":"A Tepljakov","year":"2018","unstructured":"Tepljakov A, Alagoz BB, Yeroglu C, Gonzalez E, Hosseinnia SH, Petlenkov E (2018) Fopid controllers and their industrial applications: A survey of recent results. IFAC-PaperOnLine 51:25\u201330. https:\/\/doi.org\/10.1016\/j.ifacol.2018.06.098","journal-title":"IFAC-PaperOnLine"},{"key":"1187_CR18","doi-asserted-by":"publisher","first-page":"267","DOI":"10.3390\/fractalfract5040267","volume":"5","author":"JD Col\u00edn-Cervantes","year":"2021","unstructured":"Col\u00edn-Cervantes JD, S\u00e1nchez-L\u00f3pez C, Ochoa-Montiel R, Torres-Mu\u00f1oz D, Hern\u00e1ndez-Mej\u00eda CM, S\u00e1nchez-Gaspariano LA, Gonz\u00e1lez-Hern\u00e1ndez HG (2021) Rational approximations of arbitrary order: A survey. Fractal Fract 5:267. https:\/\/doi.org\/10.3390\/fractalfract5040267","journal-title":"Fractal Fract"},{"issue":"8","key":"1187_CR19","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1049\/iet-cta.2010.0332","volume":"5","author":"D Valerio","year":"2011","unstructured":"Valerio D, Costa JS (2011) Introduction to single-input, single output fractional control. IET Control Theory Appl 5(8):1033\u20131057. https:\/\/doi.org\/10.1049\/iet-cta.2010.0332","journal-title":"IET Control Theory Appl"},{"key":"1187_CR20","doi-asserted-by":"crossref","unstructured":"Maiti D, Konar A (2008) Approximation of a fractional order system by an integer order model using particle swarm optimization technique. arXiv arXiv:0811.0077","DOI":"10.1109\/TENCON.2008.4766861"},{"key":"1187_CR21","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1177\/0959651808102594","volume":"223","author":"M Tavakoli-Kakhki","year":"2009","unstructured":"Tavakoli-Kakhki M, Haeri M (2009) Model reduction in commensurate fractional-order linear systems. Proc Inst Mech Eng Part I J Syst Control Eng 223:493\u2013505. https:\/\/doi.org\/10.1177\/0959651808102594","journal-title":"Proc Inst Mech Eng Part I J Syst Control Eng"},{"key":"1187_CR22","doi-asserted-by":"publisher","unstructured":"Khanra M, Pal J, Biswas K (2013) Reduced order approximation of mimo fractional order systems. IEEE J. Emerg. Sel. Top. Circuits Syst. 3, 451\u2013458 https:\/\/doi.org\/10.1109\/JETCAS.2013.2279231","DOI":"10.1109\/JETCAS.2013.2279231"},{"key":"1187_CR23","doi-asserted-by":"publisher","unstructured":"Jagodzinski J, Lampasiak M (2024) Pade approximation method for fractional-order systems. In: Proceedings of the 2024 25th International Carpathian Control Conference (ICCC), Krynica Zdroj, Poland, pp. 1\u20136 . https:\/\/doi.org\/10.1109\/ICCC.2024.1234567","DOI":"10.1109\/ICCC.2024.1234567"},{"key":"1187_CR24","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1109\/JAS.2019.1911720","volume":"6","author":"Z Gao","year":"2019","unstructured":"Gao Z (2019) Stable model order reduction method for fractional-order systems based on unsymmetric lanczos algorithm. IEEE\/CAA J Autom Sin 6:485\u2013492. https:\/\/doi.org\/10.1109\/JAS.2019.1911720","journal-title":"IEEE\/CAA J Autom Sin"},{"key":"1187_CR25","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1080\/00207721.2014.924260","volume":"46","author":"YL Jiang","year":"2015","unstructured":"Jiang YL, Xiao ZH (2015) Arnoldi-based model reduction for fractional order linear systems. Int J Syst Sci 46:1411\u20131420. https:\/\/doi.org\/10.1080\/00207721.2014.924260","journal-title":"Int J Syst Sci"},{"key":"1187_CR26","doi-asserted-by":"publisher","first-page":"536","DOI":"10.1016\/j.ifacol.2018.10.002","volume":"51","author":"M Rydel","year":"2018","unstructured":"Rydel M, Stanislawski R, Latawiec KJ, Galek M (2018) Model order reduction of commensurate linear discrete-time fractional-order systems. IFAC-PaperOnLine 51:536\u2013541. https:\/\/doi.org\/10.1016\/j.ifacol.2018.10.002","journal-title":"IFAC-PaperOnLine"},{"key":"1187_CR27","doi-asserted-by":"publisher","unstructured":"Sarkar P, Shekh RR, Iqbal A (2016) A unified approach for reduced order modeling of fractional order system in delta domain. In: Proceedings of the 2016 International Automatic Control Conference (CACS), Taichung, Taiwan, pp. 257\u2013262 . https:\/\/doi.org\/10.1109\/CACS.2016.1234567","DOI":"10.1109\/CACS.2016.1234567"},{"key":"1187_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.ifacsc.2024.100240","volume":"27","author":"A Mondal","year":"2024","unstructured":"Mondal A, Ganguli S, Sarkar P (2024) Reduced order modeling of delta operator systems by optimal frequency fitting approach. IFAC J Syst Control 27:100240. https:\/\/doi.org\/10.1016\/j.ifacsc.2024.100240","journal-title":"IFAC J Syst Control"},{"key":"1187_CR29","doi-asserted-by":"publisher","first-page":"3008","DOI":"10.1016\/j.jfranklin.2017.03.007","volume":"354","author":"R Stanislawski","year":"2017","unstructured":"Stanislawski R, Rydel M, Latawiec KJ (2017) Modeling of discrete-time fractional-order state space systems using the balanced truncation method. J Frankl Inst 354:3008\u20133020. https:\/\/doi.org\/10.1016\/j.jfranklin.2017.03.007","journal-title":"J Frankl Inst"},{"key":"1187_CR30","doi-asserted-by":"publisher","first-page":"143","DOI":"10.3182\/20130703-3-FR-4038.00031","volume":"46","author":"R Garrappa","year":"2013","unstructured":"Garrappa R, Maione G (2013) Model order reduction on krylov subspaces for fractional linear systems. IFAC Proc 46:143\u2013148. https:\/\/doi.org\/10.3182\/20130703-3-FR-4038.00031","journal-title":"IFAC Proc"},{"key":"1187_CR31","doi-asserted-by":"publisher","unstructured":"Verma SK, Nagar SK (2016) Approximation and order reduction of fractional order siso system. In: Proceedings of the 2016 IEEE Annual India Conference (INDICON), Bangalore, India, pp. 1\u20136 . https:\/\/doi.org\/10.1109\/INDICON.2016.1234567","DOI":"10.1109\/INDICON.2016.1234567"},{"key":"1187_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/03081079.2022.1234567","volume":"51","author":"P Badri","year":"2022","unstructured":"Badri P, Sojoodi M (2022) Lmi-based robust stability and stabilization analysis of fractional-order interval systems with time-varying delay. Int J Gen Syst 51:1\u201326. https:\/\/doi.org\/10.1080\/03081079.2022.1234567","journal-title":"Int J Gen Syst"},{"key":"1187_CR33","doi-asserted-by":"publisher","first-page":"8572","DOI":"10.1109\/TIE.2020.1234567","volume":"68","author":"UM Al-Saggaf","year":"2020","unstructured":"Al-Saggaf UM, Mansouri R, Bettayeb M, Mehedi IM, Munawar K (2020) Robustness improvement of the fractional-order ladrc scheme for integer high-order system. IEEE Trans Ind Electron 68:8572\u20138581. https:\/\/doi.org\/10.1109\/TIE.2020.1234567","journal-title":"IEEE Trans Ind Electron"},{"key":"1187_CR34","unstructured":"Goldberg DE (1989) Genetic Algorithms in Search, Optimization and Machine Learning, 1st edn. Addison-Wesley Longman Publishing Co., Inc, Boston, MA"},{"key":"1187_CR35","doi-asserted-by":"crossref","unstructured":"Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceeding IEEE International Conference on Neural Networks, vol. 4. Perth, WA, Australia, pp. 1942\u20131948","DOI":"10.1109\/ICNN.1995.488968"},{"key":"1187_CR36","volume-title":"Nature-inspired metaheuristic algorithms","author":"X-S Yang","year":"2010","unstructured":"Yang X-S (2010) Nature-inspired metaheuristic algorithms. Luniver Press, Frome"},{"key":"1187_CR37","doi-asserted-by":"crossref","unstructured":"Yang XS, Deb S (2009) Cuckoo search via l\u00e9vy flights. In: Proceeding 2009 World Congress on Nature Biologically Inspired Computing (NaBIC), pp. 210\u2013214. IEEE, Coimbatore, India","DOI":"10.1109\/NABIC.2009.5393690"},{"issue":"1","key":"1187_CR38","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1109\/3477.484436","volume":"26","author":"M Dorigo","year":"1996","unstructured":"Dorigo M, Maniezzo V, Colorni A (1996) Ant system: Optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B (Cybern) 26(1):29\u201341","journal-title":"IEEE Trans Syst Man Cybern Part B (Cybern)"},{"issue":"2","key":"1187_CR39","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.advengsoft.2005.04.005","volume":"37","author":"OK Erol","year":"2006","unstructured":"Erol OK, Eksin I (2006) A new optimization method: Big bang-big crunch. Adv Eng Softw 37(2):106\u2013111","journal-title":"Adv Eng Softw"},{"issue":"2","key":"1187_CR40","doi-asserted-by":"publisher","first-page":"1617","DOI":"10.1109\/TIE.2016.2618783","volume":"64","author":"ABH Adamou-Mitiche","year":"2017","unstructured":"Adamou-Mitiche ABH, Mitiche L (2017) Multivariable systems model reduction based on the dominant modes and genetic algorithm. IEEE Trans Industr Electron 64(2):1617\u20131619","journal-title":"IEEE Trans Industr Electron"},{"key":"1187_CR41","doi-asserted-by":"crossref","unstructured":"Juneja M, Nagar S (2015) Comparative study of model order reduction using combination of pso with conventional reduction techniques. In: Proceeding 2015 International Conference on Industrial Instrumentation and Control (ICIC), pp. 406\u2013411. IEEE, Pune, India","DOI":"10.1109\/IIC.2015.7150776"},{"issue":"1","key":"1187_CR42","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1080\/21642583.2013.804463","volume":"1","author":"S Desai","year":"2013","unstructured":"Desai S, Prasad R (2013) A new approach to order reduction using stability equation and big bang big crunch optimization. Syst Sci Control Eng 1(1):20\u201327","journal-title":"Syst Sci Control Eng"},{"issue":"15","key":"1187_CR43","doi-asserted-by":"publisher","first-page":"7225","DOI":"10.1016\/j.apm.2016.03.006","volume":"40","author":"S Biradar","year":"2016","unstructured":"Biradar S, Hote YV, Saxena S (2016) Reduced-order modeling of linear time invariant systems using big bang big crunch optimization and time moment matching method. Appl Math Model 40(15):7225\u20137244","journal-title":"Appl Math Model"},{"issue":"16","key":"1187_CR44","doi-asserted-by":"publisher","first-page":"8016","DOI":"10.1016\/j.apm.2013.02.052","volume":"37","author":"S Desai","year":"2013","unstructured":"Desai S, Prasad R (2013) A novel order diminution of lti systems using big bang big crunch optimization and routh approximation. Appl Math Model 37(16):8016\u20138028","journal-title":"Appl Math Model"},{"issue":"1","key":"1187_CR45","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1016\/j.jfranklin.2013.09.005","volume":"351","author":"W Krajewski","year":"2014","unstructured":"Krajewski W, Viaro U (2014) A method for the integer-order approximation of fractional-order systems. J Franklin Inst 351(1):555\u2013564","journal-title":"J Franklin Inst"},{"issue":"3","key":"1187_CR46","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1080\/00207721.2012.723055","volume":"45","author":"J Shen","year":"2014","unstructured":"Shen J, Lam J (2014) State feedback h$$\\infty $$ control of commensurate fractional-order systems. Int J Syst Sci 45(3):363\u2013372","journal-title":"Int J Syst Sci"},{"issue":"7","key":"1187_CR47","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1016\/S0005-1098(00)00011-X","volume":"36","author":"H-F Raynaud","year":"2000","unstructured":"Raynaud H-F, Zerga\u0131noh A (2000) State-space representation for fractional order controllers. Automatica 36(7):1017\u20131021. https:\/\/doi.org\/10.1016\/S0005-1098(00)00011-X","journal-title":"Automatica"},{"issue":"7","key":"1187_CR48","doi-asserted-by":"publisher","first-page":"4317","DOI":"10.1137\/09075304","volume":"48","author":"M Tavakoli-Kakhki","year":"2010","unstructured":"Tavakoli-Kakhki M, Haeri M (2010) The minimal state space realization for a class of fractional order transfer functions. SIAM J Control Optim 48(7):4317\u20134326. https:\/\/doi.org\/10.1137\/09075304","journal-title":"SIAM J Control Optim"},{"issue":"5","key":"1187_CR49","doi-asserted-by":"publisher","first-page":"1099","DOI":"10.1109\/TCSI.2010.2090568","volume":"58","author":"M Tavakoli-Kakhki","year":"2010","unstructured":"Tavakoli-Kakhki M, Haeri M, Tavazoei MS (2010) Notes on the state space realizations of rational order transfer functions. IEEE Trans Circ Syst I Regul Pap 58(5):1099\u20131108. https:\/\/doi.org\/10.1109\/TCSI.2010.2090568","journal-title":"IEEE Trans Circ Syst I Regul Pap"},{"key":"1187_CR50","doi-asserted-by":"crossref","unstructured":"Bhaskaran T, Chen Y, Bohannan G (2007) Practical tuning of fractional order proportional and integral controller (ii): Experiments. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 4806, pp. 1371\u20131384","DOI":"10.1115\/DETC2007-34910"},{"key":"1187_CR51","doi-asserted-by":"crossref","unstructured":"Sreekumar HK, Ullmann R, Sicklinger S, Langer SC (2021) Efficient krylov subspace techniques for model order reduction of automotive structures in vibroacoustic applications. Model Reduction of Complex Dynamical Systems, 259\u2013282","DOI":"10.1007\/978-3-030-72983-7_12"},{"key":"1187_CR52","doi-asserted-by":"publisher","unstructured":"Freund RW (2022) Electronic circuit simulation and the development of new krylov-subspace methods. In: Novel Mathematics Inspired by Industrial Challenges, pp. 29\u201355. Springer, Cham, Switzerland . https:\/\/doi.org\/10.1007\/978-3-030-96173-2_2","DOI":"10.1007\/978-3-030-96173-2_2"},{"issue":"5","key":"1187_CR53","doi-asserted-by":"publisher","first-page":"70039","DOI":"10.1002\/nla.70039","volume":"32","author":"H Barkouki","year":"2025","unstructured":"Barkouki H, Jbilou K (2025) A rational krylov method for large-scale linear multidimensional dynamical systems. Numer Linear Algebr Appl 32(5):70039","journal-title":"Numer Linear Algebr Appl"},{"key":"1187_CR54","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2020.107135","volume":"148","author":"C Sittl","year":"2021","unstructured":"Sittl C, Marburg S, Wagner M (2021) Application of a krylov subspace method for an efficient solution of acoustic transfer functions. Mech Syst Signal Process 148:107135","journal-title":"Mech Syst Signal Process"},{"issue":"1","key":"1187_CR55","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s11075-023-01648-0","volume":"96","author":"F Bouyghf","year":"2024","unstructured":"Bouyghf F, Messaoudi A, Sadok H (2024) A unified approach to krylov subspace methods for solving linear systems. Numer Algor 96(1):305\u2013332","journal-title":"Numer Algor"},{"key":"1187_CR56","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijmecsci.2022.107171","volume":"221","author":"Y Xing","year":"2022","unstructured":"Xing Y, Gen L, Yuan Y (2022) A review of the analytical solution methods for the eigenvalue problems of rectangular plates. Int J Mech Sci 221:107171","journal-title":"Int J Mech Sci"},{"key":"1187_CR57","first-page":"1","volume":"60","author":"J Liu","year":"2022","unstructured":"Liu J, Ren Z, Xiao X, Tang J, Lin P (2022) Accelerating the frequency domain controlled-source electromagnetic data inversion using rational krylov subspace algorithm. IEEE Trans Geosci Remote Sens 60:1\u201312","journal-title":"IEEE Trans Geosci Remote Sens"},{"key":"1187_CR58","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.apnum.2021.04.007","volume":"166","author":"L Reichel","year":"2021","unstructured":"Reichel L, Ugwu UO (2021) Tensor krylov subspace methods with an invertible linear transform product applied to image processing. Appl Numer Math 166:186\u2013207","journal-title":"Appl Numer Math"},{"issue":"2","key":"1187_CR59","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1137\/22M1479853","volume":"44","author":"T Chen","year":"2023","unstructured":"Chen T, Greenbaum A, Musco C, Musco C (2023) Low-memory krylov subspace methods for optimal rational matrix function approximation. SIAM J Matrix Anal Appl 44(2):670\u2013692","journal-title":"SIAM J Matrix Anal Appl"},{"key":"1187_CR60","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113034","volume":"381","author":"Q-Y Hu","year":"2021","unstructured":"Hu Q-Y, Wen C, Huang T-Z, Shen Z-L, Gu X-M (2021) A variant of the power-arnoldi algorithm for computing pagerank. J Comput Appl Math 381:113034","journal-title":"J Comput Appl Math"},{"issue":"8","key":"1187_CR61","first-page":"1411","volume":"46","author":"Y-L Jiang","year":"2015","unstructured":"Jiang Y-L, Xiao Z-H (2015) Arnoldi-based model reduction for fractional order linear systems. Int J Syst Sci 46(8):1411\u20131420","journal-title":"Int J Syst Sci"},{"issue":"1","key":"1187_CR62","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s11075-023-01653-3","volume":"96","author":"A Bouhamidi","year":"2024","unstructured":"Bouhamidi A, Elbouyahyaoui L, Heyouni M (2024) The constant solution method for solving large-scale differential sylvester matrix equations with time invariant coefficients. Numer Algor 96(1):449\u2013488","journal-title":"Numer Algor"},{"issue":"2","key":"1187_CR63","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1007\/s10915-022-01808-9","volume":"91","author":"AH Bentbib","year":"2022","unstructured":"Bentbib AH, Ghomari ME, Jbilou K (2022) An extended-rational arnoldi method for large matrix exponential evaluations. J Sci Comput 91(2):36","journal-title":"J Sci Comput"},{"issue":"2","key":"1187_CR64","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1137\/22M1491241","volume":"46","author":"S Thomas","year":"2024","unstructured":"Thomas S, Carson E, Rozlo\u017en\u00edk M, Carr A, \u015awirydowicz K (2024) Iterated gauss-seidel gmres. SIAM J Sci Comput 46(2):254\u2013279","journal-title":"SIAM J Sci Comput"},{"issue":"351","key":"1187_CR65","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1090\/mcom\/3963","volume":"94","author":"T Chen","year":"2025","unstructured":"Chen T, Greenbaum A, Trogdon T (2025) Gmres, pseudospectra, and crouzeix\u2019s conjecture for shifted and scaled ginibre matrices. Math Comput 94(351):241\u2013261","journal-title":"Math Comput"},{"issue":"1","key":"1187_CR66","doi-asserted-by":"publisher","first-page":"5911499","DOI":"10.1155\/2023\/5911499","volume":"2023","author":"KK Kumar","year":"2023","unstructured":"Kumar KK, Ramarao G, Kumar PP, Nuvvula RS, Colak I, Khan B, Hossain MA (2023) Reduction of high dimensional noninteger commensurate systems based on differential evolution. Int Trans Electr Energy Syst 2023(1):5911499. https:\/\/doi.org\/10.1155\/2023\/5911499","journal-title":"Int Trans Electr Energy Syst"},{"key":"1187_CR67","doi-asserted-by":"crossref","unstructured":"Petras I (2014) Practical aspects for implementation of fractional order controllers. In: Proceedings of the 2014 15th International Carpathian Control Conference (ICCC), pp. 428\u2013431","DOI":"10.1109\/CarpathianCC.2014.6843641"},{"key":"1187_CR68","unstructured":"Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering. Elsevier Science, Slovak Republic"},{"key":"1187_CR69","volume-title":"The fractional calculus; theory and applications of differentiation and integration to arbitrary order","author":"KB Oldham","year":"1974","unstructured":"Oldham KB, Spanier J (1974) The fractional calculus; theory and applications of differentiation and integration to arbitrary order. Academic Press, New York"},{"issue":"1","key":"1187_CR70","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/356215","volume":"2013","author":"M Rivero","year":"2013","unstructured":"Rivero M, Rogosin SV, Tenreiro Machado JA, Trujillo JJ (2013) Stability of fractional order systems. Math Probl Eng 2013(1):356215. https:\/\/doi.org\/10.1155\/2013\/356215","journal-title":"Math Probl Eng"},{"key":"1187_CR71","doi-asserted-by":"publisher","DOI":"10.1016\/j.jappgeo.2022.104573","volume":"198","author":"K Lu","year":"2022","unstructured":"Lu K, Zhou J, Li X, Fan Y, Qi Z, Cao H (2022) 3d large-scale transient electromagnetic modeling using a shift-and-invert krylov subspace method. J Appl Geophys 198:104573. https:\/\/doi.org\/10.1016\/j.jappgeo.2022.104573","journal-title":"J Appl Geophys"},{"issue":"11","key":"1187_CR72","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1134\/S0965542521110099","volume":"61","author":"VP Il\u2019in","year":"2021","unstructured":"Il\u2019in VP (2021) Iterative preconditioned methods in krylov spaces: trends of the 21st century. Comput Math Math Phys 61(11):1750\u20131775. https:\/\/doi.org\/10.1134\/S0965542521110099","journal-title":"Comput Math Math Phys"},{"issue":"02","key":"1187_CR73","doi-asserted-by":"publisher","first-page":"2250041","DOI":"10.1142\/S0219876222500414","volume":"20","author":"N Kushida","year":"2023","unstructured":"Kushida N, Okuda H (2023) A new type of variable preconditioning for a generalized minimum residual scheme. Int J Comput Methods 20(02):2250041. https:\/\/doi.org\/10.1142\/S0219876222500414","journal-title":"Int J Comput Methods"},{"issue":"1","key":"1187_CR74","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1177\/10943420231207","volume":"38","author":"R Iakymchuk","year":"2024","unstructured":"Iakymchuk R, Graillat S, Aliaga JI (2024) General framework for re-assuring numerical reliability in parallel krylov solvers: A case of bi-conjugate gradient stabilized methods. Int J High Perform Comput Appl 38(1):17\u201333. https:\/\/doi.org\/10.1177\/10943420231207","journal-title":"Int J High Perform Comput Appl"},{"key":"1187_CR75","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/4074","author":"T Li","year":"2025","unstructured":"Li T, Wang Q-W (2025) Qqmr: A structure preserving quaternion quasi-minimal residual method. Math Comput. https:\/\/doi.org\/10.1090\/mcom\/4074","journal-title":"Math Comput"},{"issue":"5","key":"1187_CR76","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1007\/s41365-021-00888-w","volume":"32","author":"H-Y Wei","year":"2021","unstructured":"Wei H-Y, Briggs K, Quintanilla V, Chen Y-T (2021) Evaluation of different krylov subspace methods for simulation of the water faucet problem. Nucl Sci Tech 32(5):44. https:\/\/doi.org\/10.1007\/s41365-021-00888-w","journal-title":"Nucl Sci Tech"},{"key":"1187_CR77","doi-asserted-by":"publisher","unstructured":"Il\u2019in V, Kozlov D, Petukhov A (2021) On the minimal residual methods for solving diffusion-convection slaes. In: Journal of Physics: Conference Series, vol. 2099, p. 012005 . https:\/\/doi.org\/10.1088\/1742-6596\/2099\/1\/012005 . IOP Publishing","DOI":"10.1088\/1742-6596\/2099\/1\/012005"},{"issue":"5","key":"1187_CR78","doi-asserted-by":"publisher","first-page":"827","DOI":"10.1177\/01423312241257298","volume":"47","author":"Y-P Bu","year":"2025","unstructured":"Bu Y-P (2025) Krylov subspace model order reduction of linear dynamical systems with quadratic output. Trans Inst Meas Control 47(5):827\u2013838. https:\/\/doi.org\/10.1177\/01423312241257298","journal-title":"Trans Inst Meas Control"},{"issue":"4","key":"1187_CR79","doi-asserted-by":"publisher","first-page":"2243","DOI":"10.1137\/23M1595904","volume":"45","author":"L Burke","year":"2024","unstructured":"Burke L, G\u00fcttel S (2024) Krylov subspace recycling with randomized sketching for matrix functions. SIAM J Matrix Anal Appl 45(4):2243\u20132262. https:\/\/doi.org\/10.1137\/23M1595904","journal-title":"SIAM J Matrix Anal Appl"},{"issue":"3","key":"1187_CR80","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s11044-022-09872-7","volume":"58","author":"L Frie","year":"2023","unstructured":"Frie L, Eberhard P (2023) On shift selection for krylov subspace based model order reduction: An iterative greedy approach combined with singular value decomposition. Multibody SysDyn 58(3):231\u2013251. https:\/\/doi.org\/10.1007\/s11044-022-09872-7","journal-title":"Multibody SysDyn"},{"issue":"6","key":"1187_CR81","doi-asserted-by":"publisher","first-page":"688","DOI":"10.1016\/j.sysconle.2012.03.005","volume":"61","author":"G Flagg","year":"2012","unstructured":"Flagg G, Beattie C, Gugercin S (2012) Convergence of the iterative rational krylov algorithm. Syst Control Lett 61(6):688\u2013691","journal-title":"Syst Control Lett"},{"issue":"2","key":"1187_CR82","doi-asserted-by":"publisher","first-page":"976","DOI":"10.1137\/24M1661236","volume":"63","author":"Y Li","year":"2025","unstructured":"Li Y, Zikatanov LT, Zuo C (2025) Reduced krylov basis methods for parametric partial differential equations. SIAM J Numer Anal 63(2):976\u2013999. https:\/\/doi.org\/10.1137\/24M1661236","journal-title":"SIAM J Numer Anal"},{"issue":"1","key":"1187_CR83","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1080\/00207721.2021.1942587","volume":"53","author":"S Saxena","year":"2022","unstructured":"Saxena S, Biradar S (2022) Fractional-order imc controller for high-order system using reduced-order modelling via big-bang, big-crunch optimisation. Int J Syst Sci 53(1):168\u2013181. https:\/\/doi.org\/10.1080\/00207721.2021.1942587","journal-title":"Int J Syst Sci"},{"issue":"8","key":"1187_CR84","doi-asserted-by":"publisher","first-page":"4506","DOI":"10.1007\/s00034-023-02304-7","volume":"42","author":"BB Duddeti","year":"2023","unstructured":"Duddeti BB, Naskar AK, Subhashini K (2023) Order reduction of lti systems using balanced truncation and particle swarm optimization algorithm. Circ Syst Signal Process 42(8):4506\u20134552. https:\/\/doi.org\/10.1007\/s00034-023-02304-7","journal-title":"Circ Syst Signal Process"},{"issue":"9","key":"1187_CR85","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1007\/s00542-023-05605-8","volume":"30","author":"R Kumar","year":"2024","unstructured":"Kumar R, Sikander A (2024) Review and analysis of model order reduction techniques for high-dimensional complex systems. Microsyst Technol 30(9):1177\u20131190. https:\/\/doi.org\/10.1007\/s00542-023-05605-8","journal-title":"Microsyst Technol"},{"key":"1187_CR86","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.camwa.2023.03.009","volume":"140","author":"M Abbaszadeh","year":"2023","unstructured":"Abbaszadeh M, Azis MI, Dehghan M, Mohammadi-Arani R (2023) Reduced order model for simulation of air pollution model and application in 2d urban street canyons via the meshfree gradient smoothing method. Comput Math Appl 140:195\u2013210. https:\/\/doi.org\/10.1016\/j.camwa.2023.03.009","journal-title":"Comput Math Appl"},{"issue":"2","key":"1187_CR87","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1080\/02564602.2020.1842257","volume":"39","author":"AK Prajapati","year":"2022","unstructured":"Prajapati AK, Prasad R (2022) Model reduction using the balanced truncation method and the pad\u00e9 approximation method. IETE Tech Rev 39(2):257\u2013269. https:\/\/doi.org\/10.1080\/02564602.2020.1842257","journal-title":"IETE Tech Rev"},{"key":"1187_CR88","doi-asserted-by":"crossref","unstructured":"Matignon D (1998) Stability properties for generalized fractional differential systems. In: ESAIM: Proceedings, vol. 5, pp. 145\u2013158 . Paris, France","DOI":"10.1051\/proc:1998004"},{"key":"1187_CR89","doi-asserted-by":"crossref","unstructured":"Petr\u00e1\u0161 I (2021) Stability of fractional-order systems. Fractional-order nonlinear systems: modeling. analysis and simulation. Springer, Berlin, Heidelberg, pp 55\u2013101","DOI":"10.1007\/978-3-642-18101-6_4"},{"issue":"1","key":"1187_CR90","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s11071-009-9638-1","volume":"61","author":"I Kheirizad","year":"2010","unstructured":"Kheirizad I, Tavazoei MS, Jalali AA (2010) Stability criteria for a class of fractional order systems. Nonlinear Dyn 61(1):153\u2013161","journal-title":"Nonlinear Dyn"}],"container-title":["Evolutionary Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12065-026-01187-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12065-026-01187-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12065-026-01187-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T00:44:24Z","timestamp":1776041064000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12065-026-01187-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,13]]},"references-count":90,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["1187"],"URL":"https:\/\/doi.org\/10.1007\/s12065-026-01187-2","relation":{},"ISSN":["1864-5909","1864-5917"],"issn-type":[{"value":"1864-5909","type":"print"},{"value":"1864-5917","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,13]]},"assertion":[{"value":"9 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 April 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"All authors have been personally and actively involved in substantial work leading to the paper, and will take public responsibility for its content.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"I consent to participate in the research project and the following has been explained to me: the research may not be of direct benefit to me, my participation is completely voluntary. My right to withdraw from the study at any time without any implications to me.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"I, the undersigned, give my consent for the publication of identifiable details, which can include a photograph(s) and\/or videos and\/or case history and\/or details within the text (\u201cMaterial\u201d) to be published in the above Journal and Article.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare no conflict of interest.","order":6,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"69"}}