{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T05:30:11Z","timestamp":1773725411077,"version":"3.50.1"},"reference-count":54,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,17]],"date-time":"2021-01-17T00:00:00Z","timestamp":1610841600000},"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>A method for the advanced construction of the dynamic matrix for Model Predictive Control (MPC) algorithms with linearization is proposed in the paper. It extends numerically efficient fuzzy algorithms utilizing skillful linearization. The algorithms combine the control performance offered by the MPC algorithms with nonlinear optimization (NMPC algorithms) with the numerical efficiency of the MPC algorithms based on linear models in which the optimization problem is a standard, easy-to-solve, quadratic programming problem with linear constraints. In the researched algorithms, the free response obtained using a nonlinear process model and the future trajectory of the control signals is used to construct an advanced dynamic matrix utilizing the easy-to-obtain fuzzy model. This leads to obtaining very good prediction and control quality very close to those offered by NMPC algorithms. The proposed approach is tested in the control system of a nonlinear chemical control plant\u2014a CSTR reactor with the van de Vusse reaction.<\/jats:p>","DOI":"10.3390\/a14010025","type":"journal-article","created":{"date-parts":[[2021,1,18]],"date-time":"2021-01-18T05:17:34Z","timestamp":1610947054000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Advanced Construction of the Dynamic Matrix in Numerically Efficient Fuzzy MPC Algorithms"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6556-7919","authenticated-orcid":false,"given":"Piotr M.","family":"Marusak","sequence":"first","affiliation":[{"name":"Institute of Control and Computation Engineering, Warsaw University of Technology, Nowowiejska 15\/19, 00-665 Warszawa, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Camacho, E.F., and Bordons, C. (1999). Model Predictive Control, Springer.","DOI":"10.1007\/978-1-4471-3398-8"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Doma\u0144ski, P.D. (2020). Performance Assessment of Predictive Control\u2014A Survey. Algorithms, 13.","DOI":"10.3390\/a13040097"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"El Youssef, J., Castle, J., and Ward, W.K. (2009). A Review of Closed\u2014Loop Algorithms for Glycemic Control in the Treatment of Type 1 Diabetes. Algorithms, 2.","DOI":"10.3390\/a2010518"},{"key":"ref_4","unstructured":"Maciejowski, J.M. (2002). Predictive Control with Constraints, Prentice Hall."},{"key":"ref_5","first-page":"325","article-title":"Efficient MPC algorithms with variable trajectories of parameters weighting predicted control errors","volume":"30","author":"Nebeluk","year":"2020","journal-title":"Arch. Control Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Plamowski, S., and Kephart, R.W. (2020). The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects. Algorithms, 13.","DOI":"10.3390\/a13080178"},{"key":"ref_7","unstructured":"Rossiter, J.A. (2003). Model\u2013Based Predictive Control: A Practical Approach, CRC Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sands, T. (2019). Comparison and Interpretation Methods for Predictive Control of Mechanics. Algorithms, 12.","DOI":"10.3390\/a12110232"},{"key":"ref_9","unstructured":"Tatjewski, P. (2007). Advanced Control of Industrial Processes; Structures and Algorithms, Springer."},{"key":"ref_10","unstructured":"Blevins, T.L., McMillan, G.K., Wojsznis, W.K., and Brown, M.W. (2003). Advanced Control Unleashed, The ISA Society."},{"key":"ref_11","first-page":"133","article-title":"Cooperation of model predictive control with steady\u2013state economic optimisation","volume":"37","author":"Marusak","year":"2008","journal-title":"Control Cybern."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Marusak, P. (2020). Numerically Efficient Fuzzy MPC Algorithm with Advanced Generation of Prediction: Application to a Chemical Reactor. Algorithms, 13.","DOI":"10.3390\/a13060143"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"313","DOI":"10.2478\/amcs-2014-0023","article-title":"Disturbance modeling and state estimation for offset\u2013free predictive control with state\u2013spaced process models","volume":"24","author":"Tatjewski","year":"2014","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Abdelaal, M., and Sch\u00f6n, S. (2020). Predictive Path Following and Collision Avoidance of Autonomous Connected Vehicles. Algorithms, 13.","DOI":"10.3390\/a13030052"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1016\/S0005-1098(98)00073-9","article-title":"A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability","volume":"34","author":"Chen","year":"1998","journal-title":"Automatica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1515\/acsc-2017-0035","article-title":"Offset\u2013free nonlinear model predictive control with state\u2013space process models","volume":"27","author":"Tatjewski","year":"2017","journal-title":"Arch. Control Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/S0959-1524(01)00023-3","article-title":"Real\u2013time optimization and nonlinear model predictive control of processes governed by differential\u2013algebraic equations","volume":"12","author":"Diehl","year":"2002","journal-title":"J. Process Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1016\/j.cep.2006.06.024","article-title":"Fast reduced multiple shooting methods for nonlinear model predictive control","volume":"46","author":"Diehl","year":"2007","journal-title":"Chem. Eng. Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1016\/j.jprocont.2008.06.003","article-title":"A fast moving horizon estimation algorithm based on nonlinear programming sensitivity","volume":"18","author":"Zavala","year":"2008","journal-title":"J. Process Control"},{"key":"ref_20","unstructured":"Dominguez, L.F., and Pistikopoulos, E.N. (2010, January 1\u20133). A Novel mp-NLP Algorithm for Explicit\/Multi-parametric NMPC. Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems, Bologna, Italy."},{"key":"ref_21","unstructured":"Johansen, T.A. (2002, January 10\u201313). On multi\u2013parametric nonlinear programming and explicit nonlinear model predictive control. Proceedings of the 41st IEEE Conf Decision and Control, Las Vegas, NV, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.automatica.2003.09.021","article-title":"Approximate explicit receding horizon control of constrained nonlinear systems","volume":"40","author":"Johansen","year":"2004","journal-title":"Automatica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0098-1354(01)00739-6","article-title":"On\u2013line optimization via off\u2013line parametric optimization tools","volume":"26","author":"Pistikopoulos","year":"2002","journal-title":"Comput. Chem. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0005-1098(01)00174-1","article-title":"The explicit linear quadratic regulator for constrained systems","volume":"38","author":"Bemporad","year":"2002","journal-title":"Automatica"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bemporad, A., Borrelli, F., and Morari, M. (2000, January 28\u201330). Piecewise linear optimal controllers for hybrid systems. Proceedings of the 2000 American Control Conference, Chicago, IL, USA.","DOI":"10.1109\/ACC.2000.876688"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.isatra.2016.04.019","article-title":"Optimal partitioning of a boiler\u2013turbine unit for Fuzzy model predictive control","volume":"64","author":"Khooban","year":"2016","journal-title":"ISA Trans."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.isatra.2018.12.041","article-title":"Disturbance\u2013observer\u2013based fuzzy model predictive control for nonlinear processes with disturbances and input constraints","volume":"90","author":"Kong","year":"2019","journal-title":"ISA Trans."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.isatra.2019.01.003","article-title":"Generalized Discrete\u2013time Nonlinear Disturbance Observer Based Fuzzy Model Predictive Control for Boiler\u2013Turbine Systems","volume":"90","author":"Kong","year":"2019","journal-title":"ISA Trans."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"104364","DOI":"10.1016\/j.conengprac.2020.104364","article-title":"Robust fuzzy model predictive control for energy management systems in fuel cell vehicles","volume":"98","author":"Shen","year":"2020","journal-title":"Control Eng. Pract."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.isatra.2014.11.018","article-title":"Fuzzy modeling and predictive control of superheater steam temperature for power plant","volume":"56","author":"Wu","year":"2015","journal-title":"ISA Trans."},{"key":"ref_31","first-page":"267","article-title":"Stability analysis of nonlinear control systems with unconstrained fuzzy predictive controllers","volume":"12","author":"Marusak","year":"2002","journal-title":"Arch. Control Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2011","DOI":"10.1016\/j.ifacol.2017.08.193","article-title":"T\u2013S fuzzy model predictive speed control of electrical vehicles","volume":"50","author":"Killian","year":"2017","journal-title":"IFAC-Pap. Line"},{"key":"ref_33","first-page":"448","article-title":"Efficient model predictive control algorithm with fuzzy approximations of nonlinear models","volume":"5495","author":"Marusak","year":"2009","journal-title":"LNCS"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"\u0141awry\u0144czuk, M. (2014). Computationally Efficient Model Predictive Control Algorithms: A Neural Network Approach, Springer.","DOI":"10.1007\/978-3-319-04229-9"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/S0098-1354(98)00301-9","article-title":"Model predictive control: Past, present and future","volume":"23","author":"Morari","year":"1999","journal-title":"Comput. Chem. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.eswa.2017.06.039","article-title":"A new T\u2013S fuzzy model predictive control for nonlinear processes","volume":"88","author":"Boulkaibet","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.tsep.2018.11.010","article-title":"Adaptive neuro-fuzzy inference system (ANFIS)\u2014Based model predictive control (MPC) for carbon dioxide reforming of methane (CDRM) in a plug flow tubular reactor for hydrogen production","volume":"9","author":"Essien","year":"2019","journal-title":"Therm. Sci. Eng. Prog."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"833","DOI":"10.1515\/amcs-2015-0060","article-title":"Nonlinear state\u2013space predictive control with on\u2013line linearisation and state estimation","volume":"25","year":"2015","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1111","DOI":"10.1016\/j.asoc.2009.02.013","article-title":"Advantages of an easy to design fuzzy predictive algorithm in control systems of nonlinear chemical reactors","volume":"9","author":"Marusak","year":"2009","journal-title":"Appl. Soft Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"5826","DOI":"10.1109\/TIE.2017.2782201","article-title":"Nonlinear Monotonically Convergent Iterative Learning Control for Batch Processes","volume":"65","author":"Lu","year":"2018","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"17164","DOI":"10.1021\/acs.iecr.9b02370","article-title":"110th Anniversary: An Overview on Learning\u2013Based Model Predictive Control for Batch Processes","volume":"58","author":"Lu","year":"2019","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"6230","DOI":"10.1109\/TIE.2018.2873133","article-title":"Multipoint Iterative Learning Model Predictive Control","volume":"66","author":"Lu","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_43","first-page":"307","article-title":"Disturbance Measurement Utilization in the Efficient MPC Algorithm with Fuzzy Approximations of Nonlinear Models","volume":"7824","author":"Marusak","year":"2013","journal-title":"LNCS"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1109\/TSMC.1985.6313399","article-title":"Fuzzy identification of systems and its application to modeling and control","volume":"15","author":"Takagi","year":"1985","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Piegat, A. (2001). Fuzzy Modeling and Control, Physica\u2013Verlag.","DOI":"10.1007\/978-3-7908-1824-6"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"61","DOI":"10.3182\/20071029-2-FR-4913.00011","article-title":"Efficient fuzzy predictive algorithms with integrated economic optimization: A case study","volume":"40","author":"Marusak","year":"2007","journal-title":"IFAC Proc. Vol."},{"key":"ref_47","first-page":"396","article-title":"Easily reconfigurable analytical fuzzy predictive controllers: Actuator faults handling","volume":"5370","author":"Marusak","year":"2008","journal-title":"LNCS"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1590\/0104-6632.20190363s20180457","article-title":"A methodology to obtain analytical models that reduce the computational complexity faced in real time implementation of NMPC controllers","volume":"36","author":"Ribeiro","year":"2019","journal-title":"Braz. J. Chem. Eng."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1109\/LCSYS.2019.2937921","article-title":"A Stabilizing Sub\u2013Optimal Model Predictive Control for Quasi\u2013Linear Parameter Varying Systems","volume":"4","author":"Mate","year":"2020","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Jain, A., and Taparia, R. (2018, January 22\u201324). Laguerre function based model predictive control for van\u2013de\u2013vusse reactor. Proceedings of the 2nd IEEE Int. Conf. Power Electronics, Intelligent Control and Energy Systems, ICPEICES 2018, Delhi, India.","DOI":"10.1109\/ICPEICES.2018.8897438"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"7769","DOI":"10.1007\/s00500-018-3405-5","article-title":"A Runge\u2013Kutta neural network-based control method for nonlinear MIMO systems","volume":"23","year":"2019","journal-title":"Soft Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.1007\/s11063-019-10167-w","article-title":"A Novel Model Predictive Runge\u2013Kutta Neural Network Controller for Nonlinear MIMO Systems","volume":"51","year":"2020","journal-title":"Neural Process. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1016\/j.ifacol.2018.09.351","article-title":"Tracking error plus damping injection control of non-minimum phase processes","volume":"51","author":"Hoang","year":"2018","journal-title":"IFAC-Pap. Line"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/0005-1098(94)00150-H","article-title":"Nonlinear model\u2013based control using second\u2013order Volterra models","volume":"31","author":"Doyle","year":"1995","journal-title":"Automatica"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/1\/25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:12:10Z","timestamp":1760159530000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/1\/25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,17]]},"references-count":54,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["a14010025"],"URL":"https:\/\/doi.org\/10.3390\/a14010025","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,17]]}}}