{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:23:20Z","timestamp":1760145800873,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T00:00:00Z","timestamp":1724803200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62071123","61601125","2023J011117","JAT220258","2019J01887"],"award-info":[{"award-number":["62071123","61601125","2023J011117","JAT220258","2019J01887"]}]},{"name":"Natural Science Foundation of the Fujian Province of China","award":["62071123","61601125","2023J011117","JAT220258","2019J01887"],"award-info":[{"award-number":["62071123","61601125","2023J011117","JAT220258","2019J01887"]}]},{"name":"Fujian Province Education Hall Youth Project","award":["62071123","61601125","2023J011117","JAT220258","2019J01887"],"award-info":[{"award-number":["62071123","61601125","2023J011117","JAT220258","2019J01887"]}]},{"name":"Fujian Natural Science Foundation Project","award":["62071123","61601125","2023J011117","JAT220258","2019J01887"],"award-info":[{"award-number":["62071123","61601125","2023J011117","JAT220258","2019J01887"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To enhance control over the SMB process, this paper employs the Crank\u2013Nicolson method to develop a discrete dynamical model. This approach allows for the investigation of system stability and convergence, fundamentally addressing the sources of error. During the discretization of partial differential equations (PDEs), two main types of errors arise: intrinsic errors from the method itself and truncation errors due to derivative approximations and the discretization process. Research indicates that for the former, the iterative process remains convergent as long as the time and spatial steps are sufficiently small. Regarding truncation errors, studies have demonstrated that they exhibit second-order behavior relative to time and spatial steps. The theoretical validation shows that the iteration works effectively, and simulations confirm that the finite difference method is stable and performs well with varying SMB system parameters and controller processes. This provides a solid theoretical foundation for practical, real-time online control.<\/jats:p>","DOI":"10.3390\/axioms13090586","type":"journal-article","created":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T07:52:08Z","timestamp":1724831528000},"page":"586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability and Convergence Analysis of the Discrete Dynamical System for Simulating a Moving Bed"],"prefix":"10.3390","volume":"13","author":[{"given":"Chao-Fan","family":"Xie","sequence":"first","affiliation":[{"name":"Department of Big Data and Artificial Intelligence, Fujian Polytechnic Normal University, Fuzhou 350300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4766-4221","authenticated-orcid":false,"given":"Hong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Nondestructive Testing, Fujian Polytechnic Normal University, Fuzhou 350300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rey-Chue","family":"Hwang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, I-Shou University, Kaohsiung 84001, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.chroma.2008.10.075","article-title":"Simulated moving bed chromatography for the separation of enantiomers","volume":"1216","author":"Rajendran","year":"2009","journal-title":"J. Chromatogr. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1081\/SPM-200042081","article-title":"Simulated moving bed equipment designs","volume":"33","author":"Chin","year":"2007","journal-title":"Sep. Purif. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.jbiosc.2009.08.194","article-title":"Analysis of \u201cfocusing\u201d effect in four zone SMB (simulated moving bed) unit for separation of xylose and glucose from biomass hydrolysate","volume":"108","author":"Kim","year":"2009","journal-title":"J. Biosci. Bioeng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.jprocont.2013.11.001","article-title":"Cycle to cycle adaptive control of simulated moving bed chromatographic separation processes","volume":"24","author":"Suvarov","year":"2014","journal-title":"J. Process Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.chroma.2017.02.040","article-title":"Design of simulated moving bed for separation of fumaric acid with a little fronting phenomenon","volume":"1491","author":"Choi","year":"2017","journal-title":"J. Chromatogr. A"},{"key":"ref_6","first-page":"1672","article-title":"One-step optimization strategy in the simulated moving bed process with asynchronous movement of ports: A VariCol case study","volume":"1634","author":"Supelano","year":"2020","journal-title":"J. Chromatogr. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"462280","DOI":"10.1016\/j.chroma.2021.462280","article-title":"Optimal performance comparison of the simulated moving bed process variants based on the modulation of the length of zones and the feed concentration","volume":"1651","author":"Reinaldo","year":"2021","journal-title":"J. Chromatogr. A"},{"key":"ref_8","unstructured":"S\u00e1 Gomes, P.M.D. (2009). Advances in Simulated Moving Bed: New Operating Modes: New Design Methodologies and Product (FlexSMB-LSRE) Development. [Ph.D. Thesis, University of Porto (FEUP)]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2537","DOI":"10.1016\/S0009-2509(98)00076-1","article-title":"Computationally efficient dynamic modelling and simulation of simulated moving bed chromatographic processes with linear isotherms","volume":"53","author":"Dunnebier","year":"1998","journal-title":"Chem. Eng. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.chroma.2016.09.070","article-title":"Nonlinear model predictive control applied to the separation of praziquantel in simulated moving bed chromatography","volume":"1470","author":"Neto","year":"2016","journal-title":"J. Chromatogr. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126864","DOI":"10.1016\/j.fuel.2022.126864","article-title":"Raising the Research Octane Number using an optimized Simulated Moving Bed technology towards greater sustainability and economic return","volume":"337","author":"Muhammed","year":"2023","journal-title":"Fuel"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2391891","DOI":"10.1155\/2019\/2391891","article-title":"Model Predictive Control Method of Simulated Moving Bed Chromatographic Separation Process Based on Subspace System Identification","volume":"2019","author":"Yan","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1016\/j.compchemeng.2004.01.007","article-title":"Transient and steady-state models for simulated moving bed processes: Numerical solutions","volume":"28","author":"Leao","year":"2004","journal-title":"Comput. Chem. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ces.2018.07.004","article-title":"Predicting aerosol size distribution development in absorption columns","volume":"192","author":"Majeed","year":"2018","journal-title":"Chem. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107553","DOI":"10.1016\/j.compchemeng.2021.107553","article-title":"Development of novel flow distribution apparatus for simulated moving bed to improve degree of mixing","volume":"156","author":"Kim","year":"2022","journal-title":"Comput. Chem. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"120597","DOI":"10.1016\/j.seppur.2022.120597","article-title":"Dynamic modeling and machine learning of commercial-scale simulated moving bed chromatography for application to multi-component normal paraffin separation NSTL","volume":"288","author":"Lee","year":"2022","journal-title":"Sep. Purif. Technol."},{"key":"ref_17","first-page":"1","article-title":"Predictive control method of simulated moving bed chromatographic separation process based on piecewise affine","volume":"50","author":"Li","year":"2020","journal-title":"Int. J. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"462073","DOI":"10.1016\/j.chroma.2021.462073","article-title":"Automatic control of simulated moving bed process with deep Q-network","volume":"1647","author":"Hoon","year":"2021","journal-title":"J. Chromatogr. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1016\/S0098-1354(00)00493-2","article-title":"Repetitive model predictive control applied to a simulated moving bed chromatography system","volume":"24","author":"Natarajan","year":"2000","journal-title":"Comput. Chem. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0959-1524(01)00005-1","article-title":"Mode-based control of a simulated moving bed chromatographic process for the separation of frutose and glucose","volume":"12","author":"Klatt","year":"2002","journal-title":"J. Process Control"},{"key":"ref_21","first-page":"632","article-title":"Combination of multi-model predictive control and the wave theory for the control of simulated moving bed plants","volume":"66","author":"Carlos","year":"2017","journal-title":"J. Chem. Eng. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"108318","DOI":"10.1016\/j.asoc.2021.108318","article-title":"A long short-term memory based Quasi-Virtual Analyzer for dynamic real-time soft sensing of a Simulated Moving Bed unit","volume":"116","author":"Marrocos","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1021\/acs.iecr.1c03388","article-title":"Recovery of a Succinic, Formic, and Acetic Acid Mixture from a Model Fermentation Broth by Simulated Moving Bed Adsorption with Methanol as a Desorbent","volume":"61","author":"Santos","year":"2022","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"10103","DOI":"10.1002\/amp2.10103","article-title":"Utilization of operation data for parameter estimation of simulated moving bed chromatography","volume":"4","author":"Suzuki","year":"2022","journal-title":"J. Adv. Manuf. Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1177\/0263617418804001","article-title":"Optimizing control of adsorption separation processes based on the improved moving asymptotes algorithm","volume":"36","author":"Yang","year":"2018","journal-title":"Adsorpt. Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1979","DOI":"10.1021\/acs.iecr.9b05238","article-title":"Novel Switch Stabilizing Model Predictive Control Strategy Applied in the Control of a Simulated Moving Bed for the Separation of Bi-Naphthol Enantiomers","volume":"59","author":"Nogueira","year":"2020","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Xie, C.-F., Hong, Z., and Rey-Chue, H. (2024). Discrete Dynamic System Modeling for Simulated Moving Bed Processes. Mathematics, 12.","DOI":"10.3390\/math12101520"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Crank, J., and Nicolson, P. (1947). A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations for the Heat Conduction Type, Cambridge University Press.","DOI":"10.1017\/S0305004100023197"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/586\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:44:18Z","timestamp":1760111058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/586"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,28]]},"references-count":28,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090586"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090586","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,8,28]]}}}