{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T03:04:08Z","timestamp":1771556648818,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we discuss an improved version of the conventional PID (Proportional\u2013Integral\u2013Derivative) controller, the Dynamically Updated PID (DUPID) controller. The DUPID is a control solution which preserves the advantages of the PID controller and tends to improve them by introducing a quadratic error model in the PID control structure. The quadratic error model is constructed over a window of past error points. The objective is to use the model to give the conventional PID controller the awareness needed to battle the effects caused by the variation of the parameters. The quality of the predictions that the model is able to deliver depends on the appropriate selection of data used for its construction. In this regard, the paper discusses two algorithms, named 1D (one dimensional) and 2D (two dimensional) DUPID. Appropriate to their names, the former selects data based on one coordinate, whereas the latter selects the data based on two coordinates. Both these versions of the DUPID controller are compared to the conventional PID controller with respect to their capabilities of controlling a Continuous Stirred Tank Reactor (CSTR) system with varying parameters in three different scenarios. As a quantifying measure of the control performance, the integral of absolute error (IAE) metric is used. The results from the performed simulations indicated that the two versions of the DUPID controller improved the control performance of the conventional PID controller in all scenarios.<\/jats:p>","DOI":"10.3390\/a14020031","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T09:49:21Z","timestamp":1611222561000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Quadratic Model-Based Dynamically Updated PID Control of CSTR System with Varying Parameters"],"prefix":"10.3390","volume":"14","author":[{"given":"Dushko","family":"Stavrov","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering and Information Technology, Ss. Cyril and Methodius University in Skopje, Rugjer Boshkovic br. 19, 1000 Skopje, North Macedonia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0708-4812","authenticated-orcid":false,"given":"Gorjan","family":"Nadzinski","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Information Technology, Ss. Cyril and Methodius University in Skopje, Rugjer Boshkovic br. 19, 1000 Skopje, North Macedonia"}]},{"given":"Stojche","family":"Deskovski","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Information Technology, Ss. Cyril and Methodius University in Skopje, Rugjer Boshkovic br. 19, 1000 Skopje, North Macedonia"}]},{"given":"Mile","family":"Stankovski","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Information Technology, Ss. Cyril and Methodius University in Skopje, Rugjer Boshkovic br. 19, 1000 Skopje, North Macedonia"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,21]]},"reference":[{"key":"ref_1","unstructured":"\u00c5str\u00f6m, K.J., H\u00e4gglund, T., and Astrom, K.J. (2021, January 19). Advanced PID Control. vol. 461. ISA-The Instrumentation, Systems, and Automation Society Research Triangle, Available online: https:\/\/www.researchgate.net\/publication\/3207690_Advanced_PID_Control_-_Book_Review."},{"key":"ref_2","unstructured":"Rotach, V.Y. (2008). Teoriya Avtomaticheskogo Upravleniya: Uchebnik dlya VUZov (Automated Control Theory: University Textbook), MEI Publishing House. [5th ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2014\/507142","article-title":"Adaptive PID Controller Using RLS for SISO Stable and Unstable Systems","volume":"2014","author":"Fahmy","year":"2014","journal-title":"Adv. Power Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1115\/1.2899060","article-title":"Optimum Settings for Automatic Controllers","volume":"115","author":"Ziegler","year":"1993","journal-title":"J. Dyn. Syst. Meas. Control."},{"key":"ref_5","unstructured":"Haykin, S.S. (2008). Adaptive Filter Theory, Pearson Education India."},{"key":"ref_6","unstructured":"\u00c5str\u00f6m, K.J., and Wittenmark, B. (2013). Adaptive Control, Courier Corporation."},{"key":"ref_7","unstructured":"Tian, B., Su, H., and Chu, J. (July, January 26). An optimal self-tuning PID controller considering parameter estimation uncertainty. Proceedings of the 3rd World Congress on Intelligent Control and Automation (Cat. No.00EX393), Hefei, China."},{"key":"ref_8","unstructured":"Mart\u00ednez, J.I.O., Paz Ramos, M., Garibo, S., and Luna-Ortega, C. (2009, January 10\u201313). PI controller with dynamic gains calculation to comply with time specs in presence of parametric disturbances. Proceedings of the 6th International Conference Electrical Engineering, Computing Science and Automatic Control, Toluca, Mexico."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Xiao, S., Li, Y., and Liu, J. (2012, January 20\u201324). A model reference adaptive PID control for electromagnetic actuated micro-positioning stage. Proceedings of the 2012 IEEE International Conference on Automation Science and Engineering (CASE), Seoul, Korea. Available online: http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.721.2583&rep=rep1&type=pdf.","DOI":"10.1109\/CoASE.2012.6386390"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1016\/j.automatica.2014.02.042","article-title":"Robust control of uncertain systems: Classical results and recent developments","volume":"50","author":"Petersen","year":"2014","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Vilanova, R., Alfaro, V.M., and Arrieta, O. (2012). Robustness in PID Control. PID Control in the Third Millennium, Springer.","DOI":"10.1007\/978-1-4471-2425-2"},{"key":"ref_12","unstructured":"Morari, M., and Zafiriou, E. (2021, January 19). Robust Process Control, Available online: https:\/\/cse.sc.edu\/~gatzke\/cache\/morari-zafiriou-PI-chapters1-3.pdf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1002\/aic.690440112","article-title":"PID controller tuning for desired closed-loop responses for SI\/SO systems","volume":"44","author":"Lee","year":"1998","journal-title":"AIChE J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0959-1524(00)00057-3","article-title":"Robust PID controller design via LMI approach","volume":"12","author":"Ge","year":"2002","journal-title":"J. Process. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6416","DOI":"10.3182\/20080706-5-KR-1001.01082","article-title":"A Software Tool for Robust PID Design","volume":"41","author":"Garpinger","year":"2008","journal-title":"IFAC Proc. Vol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1109\/TCST.2016.2562581","article-title":"Robust PID Design Based on QFT and Convex\u2013Concave Optimization","volume":"25","author":"Mercader","year":"2016","journal-title":"IEEE Trans. Control. Syst. Technol."},{"key":"ref_17","unstructured":"Stavrov, D., Nadzinski, G., Stojanovski, G., and Deskovski, S. (2018, January 20\u201322). Performance analysis of DUPID control over CSTR sys-tem with varying parameters. Proceedings of the 14th International Conference ETAI, Struga, Macedonia."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ifacol.2015.11.072","article-title":"Handling Disturbances in Modifier Adaptation with Quadratic Approximation. The research leading to these results was funded by the ERC Ad-vanced Investigator Grant MOBOCON under the grant agreement No. 291458","volume":"48","author":"Wenzel","year":"2015","journal-title":"IFAC-PapersOnLine"},{"key":"ref_19","unstructured":"Wenzel, S., Paulen, R., and Engell, S. (2021, January 19). Quadratic Approximation in Price-Based Coordination of Constrained Systems-of-Systems. Focapo\/cpc. Available online: http:\/\/folk.ntnu.no\/skoge\/prost\/proceedings\/focapo-cpc-2017\/FOCAPO-CPC%202017%20Contributed%20Papers\/92_CPC_Contributed.pdf."},{"key":"ref_20","unstructured":"Seborg, D.E., Mellichamp, D.A., Edgar, T.F., and Doyle, F.J. (2010). Process Dynamics and Control, John Wiley & Sons."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"O\u2019Dwyer, A. (2009). Handbook of PI and PID Controller Tuning Rules, Imperial College Press.","DOI":"10.1142\/9781848162433"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Brunton, S.L., and Kutz, J.N. (2019). Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge University Press. [1st ed.].","DOI":"10.1017\/9781108380690"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1002\/rnc.1048","article-title":"Linear PI control of batch exothermic reactors with temperature measurement","volume":"16","author":"Alvarez","year":"2006","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S1385-8947(99)00108-4","article-title":"Control of nonisothermal CSTR with time varying parameters via dy-namic neural network control (DNNC)","volume":"76","author":"Nikravesh","year":"2000","journal-title":"Chem. Eng. J."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bucz, \u0160., and Koz\u00e1kov\u00e1, A. (2018). Advanced methods of PID controller tuning for specified performance. PID Control Ind. Process, 73\u2013119. Available online: https:\/\/www.intechopen.com\/books\/pid-control-for-industrial-processes\/advanced-methods-of-pid-controller-tuning-for-specified-performance.","DOI":"10.5772\/intechopen.76069"},{"key":"ref_26","unstructured":"Yu, C.-C. (2006). Autotuning of PID Controllers: A Relay Feedback Approach, Springer Science & Business Media."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1002\/rnc.579","article-title":"Global stabilization of chemical reactors with classical PI control","volume":"11","year":"2001","journal-title":"Int. J. Robust Nonlinear Control IFAC Affil. J."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/2\/31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:26Z","timestamp":1760159606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/2\/31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,21]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["a14020031"],"URL":"https:\/\/doi.org\/10.3390\/a14020031","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,21]]}}}