{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T11:55:13Z","timestamp":1773921313161,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,6,7]],"date-time":"2020-06-07T00:00:00Z","timestamp":1591488000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["ID\/CEC\/00127\/2019"],"award-info":[{"award-number":["ID\/CEC\/00127\/2019"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>The present paper proposes an approach for the development of a non-linear model-based predictive controller (NMPC) using a non-linear process model based on Artificial Neural Networks (ANNs). This work exploits recent trends on ANN literature using a TensorFlow implementation and shows how they can be efficiently used as support for closed-loop control systems. Furthermore, it evaluates how the generalization capability problems of neural networks can be efficiently overcome when the model that supports the control algorithm is used outside of its initial training conditions. The process\u2019s transient response performance and steady-state error are parameters under focus and will be evaluated using a MATLAB\u2019s Simulink implementation of a Coupled Tank Liquid Level controller and a Yeast Fermentation Reaction Temperature controller, two well-known benchmark systems for non-linear control problems.<\/jats:p>","DOI":"10.3390\/app10113958","type":"journal-article","created":{"date-parts":[[2020,6,9]],"date-time":"2020-06-09T04:19:39Z","timestamp":1591676379000},"page":"3958","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Model Predictive Control of Non-Linear Systems Using Tensor Flow-Based Models"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7925-7869","authenticated-orcid":false,"given":"R\u00f3mulo","family":"Ant\u00e3o","sequence":"first","affiliation":[{"name":"IEETA, Institute of Electronics and Informatics Engineering of Aveiro University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2572-3681","authenticated-orcid":false,"given":"Jos\u00e9","family":"Antunes","sequence":"additional","affiliation":[{"name":"DETI, Department of Electronics, Telecommunications and Informatics University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2383-9370","authenticated-orcid":false,"given":"Alexandre","family":"Mota","sequence":"additional","affiliation":[{"name":"DETI, Department of Electronics, Telecommunications and Informatics University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1363-7794","authenticated-orcid":false,"given":"Rui","family":"Escadas Martins","sequence":"additional","affiliation":[{"name":"IEETA, Institute of Electronics and Informatics Engineering of Aveiro University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Camacho, E.F., and Bordons, C. (2007). Model Predictive Control, Springer. [2nd ed.].","DOI":"10.1007\/978-0-85729-398-5"},{"key":"ref_2","unstructured":"Cutler, C.R., and Ramaker, B.L. (1980, January 13\u201315). Dynamic matrix control\u2014A computer control algorithm. Proceedings of the Joint Automatic Control Conference, San Francisco, CA, USA."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/0005-1098(87)90087-2","article-title":"Generalized predictive control\u2014Part I. The basic algorithm","volume":"23","author":"Clarke","year":"1987","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s12555-011-0300-6","article-title":"Model predictive control: Review of the three decades of development","volume":"9","author":"Lee","year":"2011","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_5","unstructured":"Sheta, A., Braik, M., and Al-Hiary, H. (2009, January 13\u201316). Identification and Model Predictive Controller Design of the Tennessee Eastman Chemical Process Using ANN. Proceedings of the 2009 International Conference on Artificial Intelligence, Las Vegas, NV, USA."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/0009-2509(92)80263-C","article-title":"Model-predictive control of chemical processes","volume":"47","author":"Eaton","year":"1992","journal-title":"Chem. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1016\/j.jprocont.2009.01.001","article-title":"Artificial neural network based system identification and model predictive control of a flotation column","volume":"19","author":"Mohanty","year":"2009","journal-title":"J. Process Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1109\/TIE.2008.2007547","article-title":"Design and implementation of model predictive control for electrical motor drives","volume":"56","author":"Bolognani","year":"2008","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.isatra.2017.09.016","article-title":"Model predictive control for systems with fast dynamics using inverse neural models","volume":"72","author":"Stogiannos","year":"2018","journal-title":"ISA Trans."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Guillaumin, M., Verbeek, J., and Schmid, C. (2010, January 13\u201318). Multimodal semi-supervised learning for image classification. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, San Francisco, CA, USA.","DOI":"10.1109\/CVPR.2010.5540120"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1109\/TASL.2013.2244083","article-title":"Machine learning paradigms for speech recognition: An overview","volume":"21","author":"Deng","year":"2013","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Collobert, R., and Weston, J. (2008, January 5\u20139). A unified architecture for natural language processing: Deep neural networks with multitask learning. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland.","DOI":"10.1145\/1390156.1390177"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/72.80202","article-title":"Identification and control of dynamical systems using neural networks","volume":"1","author":"Narendra","year":"1990","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"James, A. (2020). Getting Started with TensorFlow Deep Learning. Deep Learning Classifiers with Memristive Networks. Modeling and Optimization in Science and Technologies, Springer.","DOI":"10.1007\/978-3-030-14524-8"},{"key":"ref_15","unstructured":"Chollet, F. (2020, June 05). Keras. Available online: https:\/\/keras.io."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Jia, Y., Shelhamer, E., Donahue, J., Karayev, S., Long, J., Girshick, R., Guadarrama, S., and Darrell, T. (2014, January 3\u20137). Caffe: Convolutional architecture for fast feature embedding. Proceedings of the ACM International Conference on Multimedia\u2014MM \u201914, Orlando, FL, USA.","DOI":"10.1145\/2647868.2654889"},{"key":"ref_17","unstructured":"Bhardwaj, S., Curtin, R.R., Edel, M., Mentekidis, Y., and Sanderson, C. (2018). Ensmallen: A flexible C++ library for efficient function optimization. arXiv."},{"key":"ref_18","unstructured":"(2020, June 05). Tensor Flow Lite. Available online: https:\/\/www.tensorflow.org\/lite\/."},{"key":"ref_19","unstructured":"Banbury, C.R., Reddi, V.J., Lam, M., Fu, W., Fazel, A., Holleman, J., Huang, X., Hurtado, R., Kanter, D., and Lokhmotov, A. (2020). Benchmarking TinyML Systems: Challenges and Direction. arXiv."},{"key":"ref_20","unstructured":"Lai, L., Suda, N., and Chandra, V. (2018). CMSIS-NN: Efficient Neural Network Kernels for Arm Cortex-M CPUs. arXiv."},{"key":"ref_21","unstructured":"Phan, R. (2020, June 05). Neural NetPlayground A MATLAB implementation of the TensorFlow Neural Network Playground, GitHub. Retrieved 5 June 2020. Available online: https:\/\/github.com\/StackOverflowMATLABchat\/NeuralNetPlayground."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"N\u00f8rgaard, M., Ravn, O., Poulsen, N., and Hansen, L.K. (2000). Neural Networks for Modelling and Control of Dynamic Systems: A Practitioner\u2019s Handbook, Springer.","DOI":"10.1007\/978-1-4471-0453-7"},{"key":"ref_23","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer. [2nd ed.]."},{"key":"ref_24","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning, The MIT Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"\u0141awry\u0144czuk, M. (2014). Computationally Efficient Model Predictive Control Algorithms\u2014A Neural Network Approach, Studies in Systems, Decision and Control, Springer International Publishing.","DOI":"10.1007\/978-3-319-04229-9"},{"key":"ref_26","unstructured":"Ogata, K. (2009). Modern Control Engineering, Prentice Hall. [5th ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ant\u00e3o, R. (2017). Type-2 Fuzzy Logic: Uncertain Systems\u2019 Modeling and Control, Springer.","DOI":"10.1007\/978-981-10-4633-9"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/S0019-0578(07)60228-6","article-title":"A Simplified Type-2 Fuzzy Logic Controller for Real-Time Control","volume":"45","author":"Wu","year":"2006","journal-title":"ISA Trans."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.cej.2006.10.015","article-title":"Model Based Control of a Yeast Fermentation Bioreactor Using Optimally Designed Artificial Neural Networks","volume":"127","author":"Nagy","year":"2007","journal-title":"Chem. Eng. J."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Luyben, W. (2007). Chemical Reactor Design and Control, AIChE Wiley.","DOI":"10.1002\/9780470134917"},{"key":"ref_31","unstructured":"Granzow, B. (2020, June 05). L-BFGS-B. Available online: https:\/\/github.com\/bgranzow\/L-BFGS-B."},{"key":"ref_32","first-page":"3527","article-title":"Disturbance Rejection in Neural Network Model Predictive Control","volume":"41","author":"Fatehi","year":"2008","journal-title":"IFAC Proc."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/10\/11\/3958\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:36:27Z","timestamp":1760175387000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/10\/11\/3958"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,7]]},"references-count":32,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["app10113958"],"URL":"https:\/\/doi.org\/10.3390\/app10113958","relation":{},"ISSN":["2076-3417"],"issn-type":[{"value":"2076-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,7]]}}}