{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T19:49:52Z","timestamp":1772567392878,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study presents a novel meta-heuristic optimization method that combines the Henry Gas Solubility Optimization (HGSO) technique with symmetric chaotic systems. By leveraging the randomness of chaotic systems, the parameters of the HGSO algorithm that require random generation are produced through chaotic processes, allowing the algorithm to exhibit chaotic behavior in its pursuit of optimal values. This innovative approach is termed Chaotic Henry Gas Solubility Optimization (CHGSO), with the primary objective of enhancing the performance of the HGSO method. The randomness of the data obtained from chaotic systems was validated using NIST-800-22 tests. The CHGSO method was applied to both 47 benchmark functions and the optimization of parameters for a PID controller utilized in the speed control of a DC motor. To evaluate the effectiveness of the proposed method, it was compared with several widely recognized algorithms in the literature, including PSO, WOA, GWO, EA, SA, and the original HGSO algorithm. The results demonstrate that the proposed method achieved the best performance in 43 of the benchmark functions, outperforming the other algorithms. In the context of controller design, the PID parameters were optimized using the error-based ITSE objective function. According to the controller responses, the proposed method has achieved the best results in the simulation studies, with a settling time of 0.035 and a rise time of 0.014 without overshooting, and in the experimental studies, with a settling time of 0.15 and a settling time of 1.4%. When the results are examined, it is observed that it has achieved successful results in the controller design problem.<\/jats:p>","DOI":"10.3390\/sym16111435","type":"journal-article","created":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T06:29:57Z","timestamp":1730183397000},"page":"1435","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Chaotic-Based Improved Henry Gas Solubility Optimization Algorithm: Application to Electric Motor Control"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2809-9896","authenticated-orcid":false,"given":"Muhammed Salih","family":"Sar\u0131kaya","sequence":"first","affiliation":[{"name":"Mechatronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, Sakarya 54050, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4757-6288","authenticated-orcid":false,"given":"Yusuf","family":"Hamida El Naser","sequence":"additional","affiliation":[{"name":"Mechatronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, Sakarya 54050, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5171-237X","authenticated-orcid":false,"given":"Sezgin","family":"Ka\u00e7ar","sequence":"additional","affiliation":[{"name":"Electrical and Electronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, Sakarya 54050, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u0130rfan","family":"Yaz\u0131c\u0131","sequence":"additional","affiliation":[{"name":"Electrical and Electronics Engineering, Faculty of Engineering, Sakarya University, Sakarya 54050, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adnan","family":"Derdiyok","sequence":"additional","affiliation":[{"name":"Mechatronics Engineering, Faculty of Technology, Sakarya University of Applied Sciences, Sakarya 54050, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wang, Z., Luo, Q., Chen, H., Zhao, J., Yao, L., Zhang, J., and Chu, F. (2024). A high\u2212accuracy intelligent fault diagnosis method for aero\u2212engine bearings with limited samples. Comput. Ind., 159.","DOI":"10.1016\/j.compind.2024.104099"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7154","DOI":"10.1016\/j.egyr.2022.05.161","article-title":"An optimal sizing framework for autonomous photovoltaic\/hydrokinetic\/hydrogen energy system considering cost, reliability and forced outage rate using horse herd optimization","volume":"8","author":"Alanazi","year":"2022","journal-title":"Energy Rep."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Fu, Y., Gao, K., Pan, Q., and Huang, M. (2024). A learning\u2212driven multi\u2212objective cooperative artificial bee colony algorithm for distributed flexible job shop scheduling problems with preventive maintenance and transportation operations. Comput. Ind. Eng., 196.","DOI":"10.1016\/j.cie.2024.110484"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/j.asoc.2019.02.003","article-title":"Fuzzy multi\u2212objective placement of renewable energy sources in distribution system with objective of loss reduction and reliability improvement using a novel hybrid method","volume":"77","author":"Nowdeh","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jahannoush, M., and Nowdeh, S.A. (2020). Optimal designing and management of a stand\u2212alone hybrid energy system using meta\u2212heuristic improved sine\u2013cosine algorithm for Recreational Center, case study for Iran country. Appl. Soft Comput., 96.","DOI":"10.1016\/j.asoc.2020.106611"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Fu, Y., Wang, Y., Gao, K., Suganthan, P.N., and Huang, M. (2024). Integrated scheduling of multi\u2212constraint open shop and vehicle routing: Mathematical model and learning\u2212driven brain storm optimization algorithm. Appl. Soft Comput., 163.","DOI":"10.1016\/j.asoc.2024.111943"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4260","DOI":"10.1109\/TSMC.2024.3376292","article-title":"Multiobjective Scheduling of Energy\u2212Efficient Stochastic Hybrid Open Shop With Brain Storm Optimization and Simulation Evaluation","volume":"54","author":"Fu","year":"2024","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00207543.2024.2356628","article-title":"Scheduling stochastic distributed flexible job shops using an multi\u2212objective evolutionary algorithm with simulation evaluation","volume":"62","author":"Fu","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1016\/j.future.2019.07.015","article-title":"Henry gas solubility optimization: A novel physics\u2212based algorithm","volume":"101","author":"Hashim","year":"2019","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mirza, A.F., Mansoor, M., and Ling, Q. (2020). A novel MPPT technique based on Henry gas solubility optimization. Energy Convers. Manag., 225.","DOI":"10.1016\/j.enconman.2020.113409"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Neggaz, N., Houssein, E.H., and Hussain, K. (2020). An efficient henry gas solubility optimization for feature selection. Expert Syst. Appl., 152.","DOI":"10.1016\/j.eswa.2020.113364"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"250","DOI":"10.5152\/electrica.2021.20088","article-title":"Implementing the Henry gas solubility optimization algorithm for optimal power system stabilizer design","volume":"21","author":"Ekinci","year":"2021","journal-title":"Electrica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"940","DOI":"10.1016\/j.net.2021.09.029","article-title":"Henry gas solubility optimization for control of a nuclear reactor: A case study","volume":"54","author":"Mousakazemi","year":"2022","journal-title":"Nucl. Eng. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1007\/s00366-020-01268-5","article-title":"A novel chaotic Henry gas solubility optimization algorithm for solving real\u2212world engineering problems","volume":"38","author":"Pholdee","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Agarwal, R., Shekhawat, N.S., and Luhach, A.K. (2021). Automated classification of soil images using chaotic Henry\u2019s gas solubility optimization: Smart agricultural system. Microprocess. Microsyst., in press.","DOI":"10.1016\/j.micpro.2021.103854"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Karasu, S., and Altan, A. (2022). Crude oil time series prediction model based on LSTM network with chaotic Henry gas solubility optimization. Energy, 242.","DOI":"10.1016\/j.energy.2021.122964"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2329","DOI":"10.1007\/s00366-021-01347-1","article-title":"Quantum Henry gas solubility optimization algorithm for global optimization","volume":"38","author":"Mohammadi","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_18","first-page":"910","article-title":"PID control for chaotic synchronization using particle swarm optimization","volume":"39","author":"Chang","year":"2009","journal-title":"Chaos"},{"key":"ref_19","first-page":"150","article-title":"Optimal tuning of fractional order PID controller for DC motor speed control using particle swarm optimization","volume":"3","author":"Rastogi","year":"2013","journal-title":"Int. J. Soft Comput. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9159","DOI":"10.1016\/j.eswa.2008.12.033","article-title":"Evolutionary algorithms based design of multivariable PID controller","volume":"36","author":"Iruthayarajan","year":"2009","journal-title":"Expert Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1109\/TSMCA.2007.914793","article-title":"A novel intelligent multiobjective simulated annealing algorithm for designing robust PID controllers","volume":"38","author":"Hung","year":"2008","journal-title":"Syst. Man Cybern. Part A Syst. Humans"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Pareek, S., Kishnani, M., and Gupta, R. (2014, January 1\u20132). Application of artificial bee colony optimization for optimal PID tuning. Proceedings of the 2014 International Conference on Advances in Engineering & Technology Research (ICAETR\u22122014), Unnao, India.","DOI":"10.1109\/ICAETR.2014.7012817"},{"key":"ref_23","first-page":"18","article-title":"Tuning PID controller for DC motor: An artificial bees optimization approach","volume":"77","year":"2013","journal-title":"Int. J. Comput. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Liao, W., Hu, Y., and Wang, H. (2014, January 10\u201312). Optimization of PID control for DC motor based on artificial bee colony algorithm. Proceedings of the 2014 International Conference on Advanced Mechatronic Systems, Kumamoto, Japan.","DOI":"10.1109\/ICAMechS.2014.6911617"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Achanta, R.K., and Pamula, V.K. (2017, January 21\u201322). DC motor speed control using PID controller tuned by jaya optimization algorithm. Proceedings of the 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI), Chennai, India.","DOI":"10.1109\/ICPCSI.2017.8391856"},{"key":"ref_26","unstructured":"Khalilpour, M., Razmjooy, N., Hosseini, H., and Moallem, P. (2011, January 25). Optimal control of DC motor using invasive weed optimization (IWO) algorithm. Proceedings of the Majlesi Conference on Electrical Engineering, Majlesi New Town, Isfahan, Iran."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103","DOI":"10.11113\/jt.v73.4254","article-title":"Gravitational search algorithm optimization for PID controller tuning in waste\u2212water treatment process","volume":"73","author":"Aziz","year":"2015","journal-title":"J. Teknol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Izci, D., Ekinci, S., Demir\u00f6ren, A., and Hedley, J. (2020, January 26\u201328). HHO algorithm based PID controller design for aircraft pitch angle control system. Proceedings of the 2020 International Congress on Human\u2212Computer Interaction, Optimization and Robotic Applications (HORA), Ankara, Turkey.","DOI":"10.1109\/HORA49412.2020.9152897"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/S1000-9361(08)60049-7","article-title":"PID controller optimization by GA and its performances on the electro\u2212hydraulic servo control system","volume":"21","author":"Elbayomy","year":"2008","journal-title":"Chin. J. Aeronaut."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Zahir, A.M., Alhady, S.S.N., Wahab, A.A.A., and Ahmad, M.F. (2020). Objective functions modification of GA optimized PID controller for brushed DC motor. Int. J. Electr. Comput. Eng., 10.","DOI":"10.11591\/ijece.v10i3.pp2426-2433"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Loucif, F., Kechida, S., and Sebbagh, A. (2020). Whale optimizer algorithm to tune PID controller for the trajectory tracking control of robot manipulator. J. Braz. Soc. Mech. Sci. Eng., 42.","DOI":"10.1007\/s40430-019-2074-3"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0022-460X(89)90873-0","article-title":"Bifurcations and chaos of a particular van der Pol\u2212duffing oscillator","volume":"132","author":"Awrejcewicz","year":"1989","journal-title":"J. Sound Vib."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1164","DOI":"10.1016\/j.physleta.2016.01.040","article-title":"Parameter estimation of Lorenz chaotic system using a hybrid swarm intelligence algorithm","volume":"380","author":"Rivera","year":"2016","journal-title":"Phys. Lett. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s11071-013-1049-7","article-title":"Controlling Rucklidge chaotic system with a single controller using linear feedback and passive control methods","volume":"75","author":"Kocamaz","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.physleta.2004.10.032","article-title":"On an optimal control design for R\u00f6ssler system","volume":"333","author":"Rafikov","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s42452-019-0776-x","article-title":"Compound difference anti\u2212synchronization between chaotic systems of integer and fractional order","volume":"1","author":"Khan","year":"2019","journal-title":"SN Appl. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF01011512","article-title":"Randomly transitional phenomena in the system governed by Duffing\u2019s equation","volume":"20","author":"Ueda","year":"1979","journal-title":"J. Stat. Phys."},{"key":"ref_38","first-page":"1","article-title":"An improved cuckoo search optimization algorithm for the problem of chaotic systems parameter estimation","volume":"2016","author":"Wang","year":"2016","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Bassham, L.E., Rukhin, A.L., Soto, J., Nechvatal, J.R., Smid, M.E., Barker, E.B., and Vo, S. (2010). A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, National Institute of Standards & Technology.","DOI":"10.6028\/NIST.SP.800-22r1a"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2593","DOI":"10.1017\/etds.2017.142","article-title":"Ergodic optimization in dynamical systems","volume":"39","author":"Jenkinson","year":"2019","journal-title":"Ergod. Theory Dyn. Syst."},{"key":"ref_41","unstructured":"Shi, Y., and Eberhart, R.C. (1998, January 4\u20139). A modified particle swarm optimizer. Proceedings of the IEEE International Conference on Evolutionary Computation, Anchorage, AK, USA."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The whale optimization algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Van Laarhoven, P.J., Aarts, E.H., van Laarhoven, P.J., and Aarts, E.H. (1987). Simulated Annealing, Springer.","DOI":"10.1007\/978-94-015-7744-1_2"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1015059928466","article-title":"Evolution strategies\u2013a comprehensive introduction","volume":"1","author":"Beyer","year":"2002","journal-title":"Nat. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1435\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:23:00Z","timestamp":1760113380000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1435"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,29]]},"references-count":45,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111435"],"URL":"https:\/\/doi.org\/10.3390\/sym16111435","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,29]]}}}