{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T21:13:19Z","timestamp":1768684399277,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T00:00:00Z","timestamp":1683072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["APVV-18-0023"],"award-info":[{"award-number":["APVV-18-0023"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["APVV-20-0046"],"award-info":[{"award-number":["APVV-20-0046"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["KEGA 012STU-4\/2021"],"award-info":[{"award-number":["KEGA 012STU-4\/2021"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["101079342"],"award-info":[{"award-number":["101079342"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Cultural and Educational Grant Agency of the Ministry of Education of Slovak Republic","award":["APVV-18-0023"],"award-info":[{"award-number":["APVV-18-0023"]}]},{"name":"Cultural and Educational Grant Agency of the Ministry of Education of Slovak Republic","award":["APVV-20-0046"],"award-info":[{"award-number":["APVV-20-0046"]}]},{"name":"Cultural and Educational Grant Agency of the Ministry of Education of Slovak Republic","award":["KEGA 012STU-4\/2021"],"award-info":[{"award-number":["KEGA 012STU-4\/2021"]}]},{"name":"Cultural and Educational Grant Agency of the Ministry of Education of Slovak Republic","award":["101079342"],"award-info":[{"award-number":["101079342"]}]},{"DOI":"10.13039\/501100000780","name":"European Union\u2019s Horizon Europe","doi-asserted-by":"publisher","award":["APVV-18-0023"],"award-info":[{"award-number":["APVV-18-0023"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union\u2019s Horizon Europe","doi-asserted-by":"publisher","award":["APVV-20-0046"],"award-info":[{"award-number":["APVV-20-0046"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union\u2019s Horizon Europe","doi-asserted-by":"publisher","award":["KEGA 012STU-4\/2021"],"award-info":[{"award-number":["KEGA 012STU-4\/2021"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000780","name":"European Union\u2019s Horizon Europe","doi-asserted-by":"publisher","award":["101079342"],"award-info":[{"award-number":["101079342"]}],"id":[{"id":"10.13039\/501100000780","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper investigates one way to reduce the computational burden of continuous-time model predictive control (MPC) laws by representing the input\/output signals and related models using B-spline functions. Such an approximation allows to implement the resulting feedback control law more efficiently, requiring less online computational effort. As a result, the proposed controller formulates the control signals as continuous polynomial spline functions. All constraints assumed over the prediction horizon are then expressed as constraints acting on the B-splines control polygon vertices. The performance of the proposed theoretical framework has been demonstrated with several real-time experiments using the well-known 2-DOF laboratory helicopter setup. The aim of the presented experiments was to track given step-like reference trajectories for pitch and yaw angles under notable parameter uncertainties. In order to suppress the influence of uncertainties, the control algorithm is implemented in an adaptive mode, equipped with the recursive least squares (RLS) estimation of model parameters and with the adaptation of stabilizing terminal set and terminal cost calculations. Thanks to the presented framework, it is possible to significantly reduce the computational burden, measured by the number of decision variables and input constrains, indicating the potential of the proposed concept for real-time applications, even when using embedded control hardware.<\/jats:p>","DOI":"10.3390\/s23094463","type":"journal-article","created":{"date-parts":[[2023,5,4]],"date-time":"2023-05-04T02:03:18Z","timestamp":1683165798000},"page":"4463","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Computationally Efficient Continuous-Time Model Predictive Control of a 2-DOF Helicopter via B-Spline Parameterization"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2867-1036","authenticated-orcid":false,"given":"Boris","family":"Roha\u013e-Ilkiv","sequence":"first","affiliation":[{"name":"Institute of Automation, Measurement and Applied Informatics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 812 31 Bratislava, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8672-9577","authenticated-orcid":false,"given":"Martin","family":"Gulan","sequence":"additional","affiliation":[{"name":"Institute of Automation, Measurement and Applied Informatics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 812 31 Bratislava, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1884-3685","authenticated-orcid":false,"given":"Peter","family":"Minar\u010d\u00edk","sequence":"additional","affiliation":[{"name":"Institute of Automation, Measurement and Applied Informatics, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, 812 31 Bratislava, Slovakia"},{"name":"Prosystemy, s.r.o. (Ltd.), 900 86 Budmerice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wang, G., Chen, J., Zhou, K., and Pang, Z. (2022). Industrial robot contouring control based on non-uniform rational B-spline curve. Symmetry, 14.","DOI":"10.3390\/sym14122533"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Usman, M., Ali, A., Tahir, A., Rahman, M.Z.U., and Khan, A.M. (2022). Efficient approach for extracting high-level B-spline features from LIDAR data for light-weight mapping. Sensors, 22.","DOI":"10.3390\/s22239168"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, G., Xu, F., Zhou, K., and Pang, Z. (2022). S-velocity profile of industrial robot based on NURBS curve and Slerp interpolation. Processes, 10.","DOI":"10.3390\/pr10112195"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Belmonte, L.M., Morales, R., Fern\u00e1ndez-Caballero, A., and Somolinos, J.A. (2016). Robust decentralized nonlinear control for a twin rotor MIMO system. Sensors, 16.","DOI":"10.5772\/64875"},{"key":"ref_5","unstructured":"Roha\u013e-Ilkiv, B., Volensk\u00fd, T., and Pavl\u00edk, L. (2007, January 11\u201314). LQ control of a laboratory helicopter model using subspace identification. Proceedings of the 16th International Conference on Process Control, \u0160trbsk\u00e9 Pleso, Slovakia."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4886","DOI":"10.1109\/ACCESS.2020.3046728","article-title":"Practical real-time implementation of a disturbance rejection control scheme for a twin-rotor helicopter system using intelligent active force control","volume":"9","author":"Abdelmaksoud","year":"2021","journal-title":"IEEE Access"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gulan, M., Minar\u010d\u00edk, P., and Liz\u00fach, M. (2019, January 11\u201314). Real-time stabilizing MPC of a 2DOF helicopter. Proceedings of the 22nd International Conference on Process Control, \u0160trbsk\u00e9 Pleso, Slovakia.","DOI":"10.1109\/PC.2019.8815035"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/0005-1098(91)90006-N","article-title":"Continuous-time generalised predictive control","volume":"27","author":"Gawthrop","year":"1991","journal-title":"Automatica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1049\/ip-cta:19990504","article-title":"Open-loop intermittent feedback control: Practical continuous-time GPC","volume":"146","author":"Ronco","year":"1999","journal-title":"IEEE Proc.-Control Theory Appl."},{"key":"ref_10","unstructured":"Wang, L. (2009). Model Predictive Control System Design and Implementation Using MATLAB, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2192","DOI":"10.1109\/TAC.2010.2053478","article-title":"On computing solutions to the continuous time constrained linear quadratic regulator","volume":"55","author":"Pannocchia","year":"2010","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1109\/TAC.2014.2324131","article-title":"Whither discrete time model predictive control?","volume":"60","author":"Pannocchia","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","unstructured":"Ferreau, H.J., Alm\u00e9r, S., Peyrl, H., Jerez, J.L., and Domahidi, A. (July, January 29). Survey of industrial applications of embedded model predictive control. Proceedings of the 15th European Control Conference, Aalborg, Denmark."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S1474-6670(17)41424-8","article-title":"One approach to continuous-time predictive control","volume":"30","year":"1997","journal-title":"IFAC Proc. Vol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Roha\u013e-Ilkiv, B., Gulan, M., and Minar\u010d\u00edk, P. (2019, January 11\u201314). Implementation of continuous-time MPC using B-spline functions. Proceedings of the 22nd International Conference on Process Control, \u0160trbsk\u00e9 Pleso, Slovakia.","DOI":"10.1109\/PC.2019.8815334"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Roha\u013e-Ilkiv, B., Gulan, M., and Minar\u010d\u00edk, P. (2020, January 11\u201317). Explicit spline-based continuous-time MPC: A study on design and performance. Proceedings of the 21st IFAC World Congress, Berlin, Germany.","DOI":"10.1016\/j.ifacol.2020.12.442"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Roha\u013e-Ilkiv, B., Gulan, M., and Minar\u010d\u00edk, P. (2021, January 1\u20134). Velocity-form spline-based continuous-time MPC: Explicit solution. Proceedings of the 23rd International Conference on Process Control, \u0160trbsk\u00e9 Pleso, Slovakia.","DOI":"10.1109\/PC52310.2021.9447544"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Betti, G., Farina, M., and Scattolini, R. (2012, January 10\u201313). An MPC algorithm for offset-free tracking of constant reference signals. Proceedings of the 51st Conference on Decision and Control, Maui, HI, USA.","DOI":"10.1109\/CDC.2012.6426758"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Pannocchia, G. (2015, January 15\u201317). Offset-free tracking MPC: A tutorial review and comparison of different formulations. Proceedings of the 14th European Control Conference, Linz, Austria.","DOI":"10.1109\/ECC.2015.7330597"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Jimoh, I.A., K\u00fc\u00e7\u00fckdemiral, I.B., Bevan, G., and Orukpe, P.E. (2020, January 15\u201318). Offset-free model predictive control: A study of different formulations with further results. Proceedings of the 28th Mediterranean Conference on Control and Automation, Saint-Raphael, France.","DOI":"10.1109\/MED48518.2020.9183056"},{"key":"ref_21","unstructured":"Navr\u00e1til, P., and Bob\u00e1l, V. (2009, January 9\u201312). Recursive Identification Algorithms Library. Proceedings of the 17th International Conference on Process Control, \u0160trbsk\u00e9 Pleso, Slovakia."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1016\/j.nonrwa.2007.12.008","article-title":"Identification of dynamic systems using multiple input\u2013single output (MISO) models","volume":"10","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"de Boor, C. (1978). A Practical Guide to Splines, Springer.","DOI":"10.1007\/978-1-4612-6333-3"},{"key":"ref_24","unstructured":"Schumaker, L.L. (1981). Spline Functions: Basic Theory, John Wiley & Sons."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Piegl, L., and Tiller, W. (1995). The NURBS Book, Springer.","DOI":"10.1007\/978-3-642-97385-7"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"H\u00f6llig, K., and H\u00f6rner, J. (2013). Approximation and Modeling with B-Splines, SIAM.","DOI":"10.1137\/1.9781611972955"},{"key":"ref_27","unstructured":"Bartels, R.H., Beatty, J.C., and Barsky, B.A. (1987). An Introduction to Splines for Use in Computer Graphic and Geometric Modeling, Morgan Kaufman Publishers, Inc."},{"key":"ref_28","unstructured":"Gopal, M. (1985). Modern Control System Theory, Wiley Eastern Limited."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.arcontrol.2004.01.006","article-title":"A note on calculation of polytopic-invariant feasible sets for linear continuous-time systems","volume":"28","year":"2004","journal-title":"Annu. Rev. Control"},{"key":"ref_30","unstructured":"Quanser (2023, January 09). Aero 2. Available online: https:\/\/www.quanser.com\/products\/aero-2\/."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s12532-014-0071-1","article-title":"qpOASES: A parametric active-set algorithm for quadratic programming","volume":"6","author":"Ferreau","year":"2014","journal-title":"Math. Program. Comput."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/9\/4463\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:28:45Z","timestamp":1760124525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/23\/9\/4463"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,3]]},"references-count":31,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["s23094463"],"URL":"https:\/\/doi.org\/10.3390\/s23094463","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,3]]}}}