{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T16:36:46Z","timestamp":1769013406639,"version":"3.49.0"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,4]],"date-time":"2019-01-04T00:00:00Z","timestamp":1546560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51839004"],"award-info":[{"award-number":["51839004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key R &amp; D Program of China","award":["2017YFC0305703"],"award-info":[{"award-number":["2017YFC0305703"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>For long-term missions in complex seas, the onboard energy resources of autonomous underwater vehicles (AUVs) are limited. Thus, energy consumption reduction is an important aspect of the study of AUVs. This paper addresses energy consumption reduction using model predictive control (MPC) based on the state space model of AUVs for trajectory tracking control. Unlike the previous approaches, which use a cost function that consists of quadratic deviations of the predicted controlled output from the reference trajectory and quadratic input changes, a term of quadratic energy (i.e., quadratic input) is introduced into the cost function in this paper. Then, the MPC control law with the new cost function is constructed, and an analysis on the effect of the quadratic energy term on the stability is given. Finally, simulation results for depth tracking control are given to demonstrate the feasibility and effectiveness of the improved MPC on energy consumption optimization for AUVs.<\/jats:p>","DOI":"10.3390\/s19010162","type":"journal-article","created":{"date-parts":[[2019,1,4]],"date-time":"2019-01-04T11:34:26Z","timestamp":1546601666000},"page":"162","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Optimization of the Energy Consumption of Depth Tracking Control Based on Model Predictive Control for Autonomous Underwater Vehicles"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5989-5174","authenticated-orcid":false,"given":"Feng","family":"Yao","sequence":"first","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Harbin Engineering University, Nangang District, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chao","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Harbin Engineering University, Nangang District, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingjun","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Harbin Engineering University, Nangang District, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yujia","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electrical Engineering, Harbin Engineering University, Nangang District, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.robot.2014.10.004","article-title":"CSurvey\u2014An autonomous optical inspection head for AUVs","volume":"67","author":"Albiez","year":"2014","journal-title":"Robot. Auton. Syst."},{"key":"ref_2","first-page":"22","article-title":"Market Prospects for AUVs","volume":"50","author":"Newman","year":"2007","journal-title":"Mar. Technol. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"9528313","DOI":"10.1155\/2018\/9528313","article-title":"Virtual Submerged Floating Operational System for Robotic Manipulation","volume":"2018","author":"Zhang","year":"2018","journal-title":"Complexity"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.oceaneng.2008.12.005","article-title":"Autonomous underwater vehicles","volume":"36","author":"Chyba","year":"2009","journal-title":"Ocean Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Medvedev, A.V., Kostenko, V.V., and Tolstonogov, A.Y. (2017, January 21\u201324). Depth control methods of variable buoyancy AUV. Proceedings of the 2017 IEEE Underwater Technology (UT), Busan, Korea.","DOI":"10.1109\/UT.2017.7890333"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1460","DOI":"10.1007\/s40815-017-0390-2","article-title":"Adaptive Fuzzy Sliding Mode Diving Control for Autonomous Underwater Vehicle with Input Constraint","volume":"20","author":"Chu","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.3390\/s150101825","article-title":"Inertial sensor self-calibration in a visually-aided navigation approach for a micro-AUV","volume":"15","author":"Boninfont","year":"2015","journal-title":"Sensors"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/MRA.2014.2385561","article-title":"Future Trends in Marine Robotics","volume":"22","author":"Zhang","year":"2015","journal-title":"Robot. Autom. Mag. IEEE"},{"key":"ref_9","first-page":"125","article-title":"Developing tendency of unmanned underwater vehicles","volume":"33","author":"Xu","year":"2011","journal-title":"Chin. J. Nat."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Eichhorn, M., Ament, C., Jacobi, M., Pfuetzenreuter, T., Karimanzira, D., Bley, M., and Wehde, H. (2018). Modular AUV System with Integrated Real-Time Water Quality Analysis. Sensors, 18.","DOI":"10.3390\/s18061837"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1017\/S0373463316000448","article-title":"Adaptive sliding mode control for depth trajectory tracking of remotely operated vehicle with thruster nonlinearity","volume":"70","author":"Chu","year":"2017","journal-title":"J. Navig."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.oceaneng.2017.09.062","article-title":"Finite-time observer based accurate tracking control of a marine vehicle with complex unknowns","volume":"145","author":"Wang","year":"2017","journal-title":"Ocean Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.3390\/s16091429","article-title":"Neural network-based self-tuning PID control for underwater vehicles","volume":"16","author":"Rodrigo","year":"2016","journal-title":"Sensors"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.oceaneng.2016.09.038","article-title":"Observer-based adaptive neural network control for a class of remotely operated vehicles","volume":"127","author":"Chu","year":"2016","journal-title":"Ocean Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.1109\/TFUZZ.2017.2737405","article-title":"Global asymptotic model-free trajectory-independent tracking control of an uncertain marine vehicle: An adaptive universe-based fuzzy control approach","volume":"26","author":"Wang","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.1016\/j.simpat.2009.04.004","article-title":"Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems","volume":"17","author":"Ho","year":"2009","journal-title":"Simul. Model. Pract. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.matcom.2015.08.021","article-title":"Modelling and simulation of a robust energy efficient AUV controller","volume":"121","author":"Sarkar","year":"2016","journal-title":"Math. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.aqpro.2015.02.074","article-title":"Energy Efficient Trajectory Tracking Controller for Underwater Applications: ARobust Approach","volume":"4","author":"Sarkar","year":"2015","journal-title":"Aquat. Procedia"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.robot.2018.05.016","article-title":"Adaptive low-level control of autonomous underwater vehicles using deep reinforcement learning","volume":"107","author":"Carlucho","year":"2018","journal-title":"Robot. Auton. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Wang, S.K., Jin, H.Z., Meng, L.W., and Li, G.C. (2016, January 27\u201329). Optimize motion energy of AUV based on LQR control strategy. Proceedings of the 2016 35th Chinese Control Conference (CCC), Chengdu, China.","DOI":"10.1109\/ChiCC.2016.7554068"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/j.conengprac.2011.12.004","article-title":"MPC: Current practice and challenges","volume":"20","author":"Darby","year":"2012","journal-title":"Control Eng. Pract."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1109\/TCYB.2015.2475376","article-title":"Hierarchical Model Predictive Image-Based Visual Servoing of Underwater Vehicles with Adaptive Neural Network Dynamic Control","volume":"46","author":"Gao","year":"2015","journal-title":"IEEE Trans. Cybern."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1109\/TII.2013.2270570","article-title":"Generalized Predictive Control with Actuator Deadband for Event-Based Approaches","volume":"10","author":"Pawlowski","year":"2014","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1109\/TCST.2016.2620061","article-title":"Event-Based GPC for Multivariable Processes: A Practical Approach with Sensor Deadband","volume":"25","author":"Pawlowski","year":"2017","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/j.enconman.2018.08.111","article-title":"A novel hybrid agent-based model predictive control for advanced building energy systems","volume":"178","author":"Sangi","year":"2018","journal-title":"Energy Convers. Manag."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.enconman.2018.10.046","article-title":"Model predictive control of indoor microclimate: Existing building stock comfort improvement","volume":"179","author":"Ryzhov","year":"2019","journal-title":"Energy Convers. Manag."},{"key":"ref_27","unstructured":"Maciejowski, J.M. (2002). Predictive Control with Constraints, Prentice-Hall."},{"key":"ref_28","unstructured":"Chen, H. (2013). Model Predictive Control, Science Press."},{"key":"ref_29","unstructured":"Wang, L.P. (2009). Model Predictive Control System Design and Implementation Using MATLAB\u00ae, Springer Science & Business Media."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/S1874-1029(13)60024-5","article-title":"Model Predictive Control\u2014Status and Challenges","volume":"39","author":"Xi","year":"2013","journal-title":"Acta Automat. Sin."},{"key":"ref_31","first-page":"191","article-title":"Model predictive control for autonomous underwater vehicle","volume":"40","author":"Budiyono","year":"2011","journal-title":"Indian J. Geo-Mar. Sci."},{"key":"ref_32","first-page":"1920","article-title":"Position and velocity control of remotely operated underwater vehicle using model predictive control","volume":"44","author":"Prasad","year":"2015","journal-title":"Indian J. Geo-Mar. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Abraham, I., and Yi, J. (2015, January 1\u20133). Model Predictive Control of buoyancy propelled autonomous underwater glider. Proceedings of the 2015 American Control Conference, Chicago, IL, USA.","DOI":"10.1109\/ACC.2015.7170893"},{"key":"ref_34","unstructured":"Steenson, L.V., Wang, L.P., Phillips, A.B., Turnock, S.R., Furlong, M.E., and Rogers, E. (2014, January 24\u201329). Experimentally verified depth regulation for AUVs using constrained model predictive control. Proceedings of the 19th World Congress of the International Federation of Automatic Control, Cape Town, South Africa."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Molero, A., Dunia, R., Cappelletto, J., and Fernandez, G. (2011, January 12\u201315). In model predictive control of remotely operated underwater vehicles. Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, USA.","DOI":"10.1109\/CDC.2011.6161447"},{"key":"ref_36","first-page":"166","article-title":"Model predictive control of a hybrid autonomous underwater vehicle with experimental verification","volume":"228","author":"Steenson","year":"2014","journal-title":"Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Yao, F., Yang, C., Liu, X., and Zhang, M.J. (2018). Experimental Evaluation on Depth Control Using Improved Model Predictive Control for Autonomous Underwater Vehicle (AUVs). Sensors, 18.","DOI":"10.3390\/s18072321"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Yang, C., Wang, Y.J., and Yao, F. (2017). Driving performance of underwater long-arm hydraulic manipulator system for small autonomous underwater vehicle and its positioning accuracy. Int. J. Adv. Rob. Syst., 14.","DOI":"10.1177\/1729881417747104"},{"key":"ref_39","unstructured":"Thor, I. (1994). Fossen. Guidance and Control of Ocean Vehicles, J. Wiley & Sons."},{"key":"ref_40","unstructured":"Zheng, D.Z. (2002). Linear System Theory, Tsinghua University Press."},{"key":"ref_41","unstructured":"Tsien, H.S., and Song, J. (2011). Engineering Cybernetics, Science Press."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.oceaneng.2018.02.007","article-title":"Adaptive fault tolerant control and thruster fault reconstruction for autonomous underwater vehicle","volume":"155","author":"Liu","year":"2018","journal-title":"Ocean Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.isatra.2012.02.003","article-title":"Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults","volume":"51","author":"Wang","year":"2012","journal-title":"ISA Trans."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/1\/162\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:23:36Z","timestamp":1760185416000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/1\/162"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,4]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1]]}},"alternative-id":["s19010162"],"URL":"https:\/\/doi.org\/10.3390\/s19010162","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1,4]]}}}