{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T02:22:39Z","timestamp":1771726959074,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T00:00:00Z","timestamp":1675382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Piezoelectric actuators (PEA) are high-precision devices used in applications requiring micrometric displacements. However, PEAs present non-linearity phenomena that introduce drawbacks at high precision applications. One of these phenomena is hysteresis, which considerably reduces their performance. The introduction of appropriate control strategies may improve the accuracy of the PEAs. This paper presents a high precision control scheme to be used at PEAs based on the model-based predictive control (MPC) scheme. In this work, the model used to feed the MPC controller has been achieved by means of artificial neural networks (ANN). This approach simplifies the obtaining of the model, since the achievement of a precise mathematical model that reproduces the dynamics of the PEA is a complex task. The presented approach has been embedded over the dSPACE control platform and has been tested over a commercial PEA, supplied by Thorlabs, conducting experiments to demonstrate improvements of the MPC. In addition, the results of the MPC controller have been compared with a proportional-integral-derivative (PID) controller. The experimental results show that the MPC control strategy achieves higher accuracy at high precision PEA applications such as tracking periodic reference signals and sudden reference change.<\/jats:p>","DOI":"10.3390\/s23031690","type":"journal-article","created":{"date-parts":[[2023,2,3]],"date-time":"2023-02-03T03:04:23Z","timestamp":1675393463000},"page":"1690","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Ultraprecise Controller for Piezoelectric Actuators Based on Deep Learning and Model Predictive Control"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7586-0588","authenticated-orcid":false,"given":"Jokin","family":"Uralde","sequence":"first","affiliation":[{"name":"Department of Systems Engineering and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9341-8251","authenticated-orcid":false,"given":"Eneko","family":"Artetxe","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4430-8088","authenticated-orcid":false,"given":"Oscar","family":"Barambones","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5256-353X","authenticated-orcid":false,"given":"Isidro","family":"Calvo","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6881-8357","authenticated-orcid":false,"given":"Pablo","family":"Fern\u00e1ndez-Bustamante","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7367-0756","authenticated-orcid":false,"given":"Imanol","family":"Martin","sequence":"additional","affiliation":[{"name":"Department of Systems Engineering and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV\/EHU), 01006 Vitoria-Gasteiz, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/TIM.2019.2950760","article-title":"An overview of piezoelectric self-sensing actuation for nanopositioning applications: Electrical circuits, displacement, and force estimation","volume":"69","author":"Liseli","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhang, P. (2010). Advanced Industrial Control Technology, Elsevier.","DOI":"10.1016\/B978-1-4377-7807-6.10010-5"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Jain, R.K., Saha, S., and Majumder, S. (2013, January 12\u201314). Development of piezoelectric actuator based compliant micro gripper for robotic peg-in-hole assembly. Proceedings of the 2013 IEEE International Conference on Robotics and Biomimetics (ROBIO), Shenzhen, China.","DOI":"10.1109\/ROBIO.2013.6739689"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ozaki, T., and Ohta, N. (2020). Power-Efficient Driver Circuit for Piezo Electric Actuator with Passive Charge Recovery. Energies, 13.","DOI":"10.3390\/en13112866"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"065027","DOI":"10.1088\/0964-1726\/19\/6\/065027","article-title":"A piezoelectric actuator with a motion-decoupling amplifier for optical disk drives","volume":"19","author":"Yang","year":"2010","journal-title":"Smart Mater. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ryndzionek, R., Sienkiewicz, \u0141., Michna, M., and Kutt, F. (2019). Design and experiments of a piezoelectric motor using three rotating mode actuators. Sensors, 19.","DOI":"10.3390\/s19235184"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.mechatronics.2008.03.002","article-title":"Adaptronic gantry machine tool with piezoelectric actuator for active error compensation of structural oscillations at the tool centre point","volume":"18","author":"Douglas","year":"2008","journal-title":"Mechatronics"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Dur\u00e1n, J.C., Escareno, J.A., Etcheverry, G., and Rakotondrabe, M. (2016). Getting started with PEAs-based flapping-wing mechanisms for micro aerial systems. Actuators, 5.","DOI":"10.3390\/act5020014"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1109\/TBME.2020.3020926","article-title":"A Direct Drive Parallel Plane Piezoelectric Needle Positioning Robot for MRI Guided Intraspinal Injection","volume":"68","author":"Meinhold","year":"2021","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_10","unstructured":"Fu, Y., Luo, J., Flewitt, A., and Milne, W. (2012). MEMS for Biomedical Applications, Elsevier."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1007\/s00542-019-04712-9","article-title":"Design and optimization of MEMS based piezoelectric actuator for drug delivery systems","volume":"26","author":"Rao","year":"2020","journal-title":"Microsyst. Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1109\/3516.891044","article-title":"Modeling piezoelectric actuators","volume":"5","author":"Adriaens","year":"2000","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liu, C., and Guo, Y. (2017). Modeling and positioning of a PZT precision drive system. Sensors, 17.","DOI":"10.3390\/s17112577"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"812","DOI":"10.3182\/20060912-3-DE-2911.00140","article-title":"Hysteresis and vibration compensation in piezoelectric actuators by integrating charge control and inverse feedforward1","volume":"39","author":"Clayton","year":"2006","journal-title":"IFAC Proc. Vol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.conengprac.2014.09.005","article-title":"Simultaneous compensation of hysteresis and creep in a single piezoelectric actuator by open-loop control for quasi-static space active optics applications","volume":"33","author":"Wang","year":"2014","journal-title":"Control Eng. Pract."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/B978-012480874-4\/50022-1","article-title":"Hysteresis in piezoelectric and ferroelectric materials","volume":"3","author":"Damjanovic","year":"2006","journal-title":"Sci. Hyster."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"17","DOI":"10.3390\/electronicmat2010002","article-title":"Hysteresis-Free Piezoresponse in Thermally Strained Ferroelectric Barium Titanate Films","volume":"2","author":"Tyunina","year":"2021","journal-title":"Electron. Mater."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1805","DOI":"10.3390\/mi6111456","article-title":"Modeling of Hysteresis in Piezoelectric Actuator Based on Segment Similarity","volume":"6","author":"Xiong","year":"2015","journal-title":"Micromachines"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23","DOI":"10.2478\/amtm-2018-0004","article-title":"Open loop control of piezoelectric tube transducer","volume":"38","author":"Stefanski","year":"2018","journal-title":"Arch. Mech. Technol. Mater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5237","DOI":"10.3182\/20110828-6-IT-1002.01063","article-title":"Hysteresis in Piezoelectric Actuators: Modeling and Compensation","volume":"44","author":"Orus","year":"2011","journal-title":"IFAC Proc. Vol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1007\/s00542-019-04437-9","article-title":"Nonlinear control design of piezoelectric actuators with micro positioning capability","volume":"27","author":"Chen","year":"2021","journal-title":"Microsyst. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"025033","DOI":"10.1088\/1361-665X\/abd894","article-title":"Feedback-feedforward control for high-speed trajectory tracking of an amplified piezoelectric actuator","volume":"30","author":"Saleem","year":"2021","journal-title":"Smart Mater. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Xiong, Y., Jia, W., Zhang, L., Zhao, Y., and Zheng, L. (2022). Feedforward Control of Piezoelectric Ceramic Actuators Based on PEA-RNN. Sensors, 22.","DOI":"10.3390\/s22145387"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"59099","DOI":"10.5772\/59099","article-title":"Recent advances in the control of piezoelectric actuators","volume":"11","author":"Chi","year":"2014","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ahmed, K., Yan, P., and Li, S. (2021). Duhem Model-Based Hysteresis Identification in Piezo-Actuated Nano-Stage Using Modified Particle Swarm Optimization. Micromachines, 12.","DOI":"10.3390\/mi12030315"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1109\/LCSYS.2021.3087102","article-title":"Robust Nonlinear Control for a Piezoelectric Actuator in a Robotic Hand Using only Position Measurements","volume":"6","author":"Flores","year":"2022","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2427","DOI":"10.1109\/9.895588","article-title":"Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis","volume":"45","author":"Su","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liu, Y., Shan, J., and Qi, N. (2014, January 24\u201327). Control system design for piezoelectric actuator based on hysteresis compensation. Proceedings of the 2014 European Control Conference (ECC), Strasbourg, France.","DOI":"10.1109\/ECC.2014.6862214"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"065204","DOI":"10.1063\/1.4953597","article-title":"Tracking control of piezoelectric actuators using a polynomial-based hysteresis model","volume":"6","author":"Gan","year":"2016","journal-title":"AIP Adv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1109\/TIE.2013.2257153","article-title":"Modeling and compensation of asymmetric Hysteresis nonlinearity for Piezoceramic actuators with a modified Prandtl-Ishlinskii model","volume":"61","author":"Gu","year":"2014","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1177\/1045389X211023583","article-title":"A rate-dependent KP modeling and direct compensation control technique for hysteresis in piezo-nanopositioning stages","volume":"33","author":"Xu","year":"2022","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zsurzsan, T.G., Andersen, M.A., Zhang, Z., and Andersen, N.A. (2015, January 9\u201312). Preisach model of hysteresis for the Piezoelectric Actuator Drive. Proceedings of the IECON 41st Annual Conference of the IEEE Industrial Electronics Society, Yokohama, Japan.","DOI":"10.1109\/IECON.2015.7392524"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1007\/s11432-009-0026-8","article-title":"Intelligent modeling and control for nonlinear systems with rate-dependent hysteresis","volume":"52","author":"Mao","year":"2009","journal-title":"Sci. China Ser. F Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1177\/0142331219869708","article-title":"Model-free robust finite-time force tracking control for piezoelectric actuators using time-delay estimation with adaptive fuzzy compensator","volume":"42","author":"Kang","year":"2020","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1109\/TMECH.2011.2128339","article-title":"A novel discrete ARMA-based model for piezoelectric actuator hysteresis","volume":"17","author":"Cao","year":"2012","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Meng, Y., Wang, X., Li, L., Huang, W., and Zhu, L. (2022). Hysteresis Modeling and Compensation of Piezoelectric Actuators Using Gaussian Process with High-Dimensional Input. Actuators, 11.","DOI":"10.3390\/act11050115"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"7717","DOI":"10.1109\/TIE.2015.2455026","article-title":"Neural-Network-Based Nonlinear Model Predictive Control for Piezoelectric Actuators","volume":"62","author":"Cheng","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Li, W., Zhang, C., Gao, W., and Zhou, M. (2020). Neural Network Self-Tuning Control for a Piezoelectric Actuator. Sensors, 20.","DOI":"10.3390\/s20123342"},{"key":"ref_39","unstructured":"Janaideh, M.A. (2013). Smart Materials-Based Actuators at the Micro\/Nano-Scale, Springer."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Du, Z., Zhou, C., Cao, Z., Wang, S., Cheng, L., and Tan, M. (2021). A neural network-based model predictive controller for displacement tracking of piezoelectric actuator with feedback delays. Int. J. Adv. Robot. Syst., 18.","DOI":"10.1177\/17298814211057698"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.precisioneng.2022.08.002","article-title":"Design of an adaptive feedforward\/feedback combined control for piezoelectric actuated micro positioning stage","volume":"78","author":"Shi","year":"2022","journal-title":"Precis. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"58984","DOI":"10.5772\/58984","article-title":"A modified Prandtl-Ishlinskii model for rate-dependent hysteresis nonlinearity using mth-power velocity damping mechanism","volume":"11","author":"Yang","year":"2014","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_43","first-page":"350","article-title":"Erratum: A neural networks based model for rate-dependent hysteresis for piezoceramic actuators (Sensors and Actuators A: Physical (2008) 143 (370\u2013376) DOI: 10.1016\/j.sna.2007.11.023)","volume":"148","author":"Ruili","year":"2008","journal-title":"Sens. Actuators A Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.ymssp.2012.10.004","article-title":"A simple fuzzy system for modelling of both rate-independent and rate-dependent hysteresis in piezoelectric actuators","volume":"36","author":"Li","year":"2013","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.rcim.2014.11.006","article-title":"Feedforward nonlinear PID control of a novel micromanipulator using Preisach hysteresis compensator","volume":"34","author":"Tang","year":"2015","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2093","DOI":"10.1007\/s00542-022-05353-1","article-title":"Non-linear hysteresis modelling of piezoelectric actuator using feedforward with PI control for micromanipulation","volume":"28","author":"Sabarianand","year":"2022","journal-title":"Microsyst. Technol."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Ma, Y., and Li, Y. (2020). Active Disturbance Compensation Based Robust Control for Speed Regulation System of Permanent Magnet Synchronous Motor. Appl. Sci., 10.","DOI":"10.3390\/app10020709"},{"key":"ref_48","unstructured":"Abidi, K., \u015eabanovi\u00e7, A., and Ye\u015filyurt, S. (2004, January 28). Sliding mode control based disturbance compensation and external force estimation for a piezoelectric actuator. Proceedings of the International Workshop on Advanced Motion Control, Kawasaki, Japan."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Chouza, A., Barambones, O., Calvo, I., and Velasco, J. (2019). Sliding mode-based robust control for piezoelectric actuators with inverse dynamics estimation. Energies, 12.","DOI":"10.3390\/en12050943"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Sve\u010dko, R., Gleich, D., and Sarja\u0161, A. (2020). The effective chattering suppression technique with adaptive super-twisted sliding mode controller based on the quasi-barrier function; an experimentation setup. Appl. Sci., 10.","DOI":"10.3390\/app10020595"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"113497","DOI":"10.1016\/j.microrel.2019.113497","article-title":"Controllable micrometer positioning design of piezoelectric actuators using a robust fuzzy eliminator","volume":"103","author":"Chen","year":"2019","journal-title":"Microelectron. Reliab."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"147077","DOI":"10.1109\/ACCESS.2020.3013935","article-title":"Linearization of Recurrent-Neural-Network- Based Models for Predictive Control of Nano-Positioning Systems Using Data-Driven Koopman Operators","volume":"8","author":"Xie","year":"2020","journal-title":"IEEE Access"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Liu, W., Cheng, L., Hou, Z.G., and Tan, M. (2015, January 1\u20133). An inversion-free model predictive control with error compensation for piezoelectric actuators. Proceedings of the American Control Conference, Chicago, IL, USA.","DOI":"10.1109\/ACC.2015.7172198"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Li, H., Zhang, Z., and Liu, Z. (2017). Application of Artificial Neural Networks for Catalysis: A Review. Catalysts, 7.","DOI":"10.3390\/catal7100306"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"118","DOI":"10.3390\/smartcities2020009","article-title":"Achieving Neuroplasticity in Artificial Neural Networks through Smart Cities","volume":"2","author":"Allam","year":"2019","journal-title":"Smart Cities"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Matrenin, P.V., Manusov, V.Z., Khalyasmaa, A.I., Antonenkov, D.V., Eroshenko, S.A., and Butusov, D.N. (2020). Improving Accuracy and Generalization Performance of Small-Size Recurrent Neural Networks Applied to Short-Term Load Forecasting. Mathematics, 8.","DOI":"10.3390\/math8122169"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.neucom.2021.10.012","article-title":"Closed-LSTM neural network based reference modification for trajectory tracking of piezoelectric actuator","volume":"467","author":"Li","year":"2022","journal-title":"Neurocomputing"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1690\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:23:09Z","timestamp":1760120589000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/3\/1690"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,3]]},"references-count":57,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["s23031690"],"URL":"https:\/\/doi.org\/10.3390\/s23031690","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,3]]}}}