{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T03:01:13Z","timestamp":1768014073155,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T00:00:00Z","timestamp":1648857600000},"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>The authors proposed an arbitrary order finite-time sliding mode control (SMC) design for a networked of uncertain higher-order nonlinear systems. A network of n+1 nodes, connected via a directed graph (with fixed topology), is considered. The nodes are considered to be uncertain in nature. A consensus error-based canonical form of the error dynamics is developed and a new arbitrary order distributed control protocol design strategy is proposed, which not only ensures the sliding mode enforcement in finite time but also confirms the finite time error dynamics stability. Rigorous stability analysis, in closed-loop, is presented, and a simulation example is given, which demonstrates the results developed in this work.<\/jats:p>","DOI":"10.3390\/s22072748","type":"journal-article","created":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T06:04:01Z","timestamp":1648965841000},"page":"2748","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Control Design for Uncertain Higher-Order Networked Nonlinear Systems via an Arbitrary Order Finite-Time Sliding Mode Control Law"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0166-4813","authenticated-orcid":false,"given":"Maryam","family":"Munir","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, HITEC University, Taxila 47080, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9051-5627","authenticated-orcid":false,"given":"Qudrat","family":"Khan","sequence":"additional","affiliation":[{"name":"Centre for Advanced Studies in Telecommunications (CAST), COMSATS University, Islamabad 45550, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8017-7006","authenticated-orcid":false,"given":"Safeer","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, COMSATS University, Islamabad 45550, Pakistan"}]},{"given":"Tayyaba Maryam","family":"Syeda","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, COMSATS University, Islamabad 45550, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1514-9185","authenticated-orcid":false,"given":"Abdullah A.","family":"Algethami","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Taif University, Taif 11099, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ren, W., and Beard, R.W. (2008). Distributed Consensus in Multi-Vehicle Cooperative Control, Springer.","DOI":"10.1007\/978-1-84800-015-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1016\/j.automatica.2007.12.010","article-title":"Average consensus problems in networks of agents with delayed communications","volume":"44","author":"Bliman","year":"2008","journal-title":"Automatica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"115","DOI":"10.3182\/20020721-6-ES-1901.00100","article-title":"Information flow and cooperative control of vehicle formations","volume":"35","author":"Fax","year":"2002","journal-title":"IFAC Proc. Vol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1109\/TAC.2007.895842","article-title":"Emergent behavior in flocks","volume":"52","author":"Cucker","year":"2007","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3575","DOI":"10.1109\/TIT.2012.2191450","article-title":"Distributed parameter estimation in sensor networks: Nonlinear observation models and imperfect communication","volume":"58","author":"Kar","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","unstructured":"Li, L., Fu, M., and Zhang, H. (2016, January 6\u20138). Consensus control for high order continuous-time agents with communication delays. Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1109\/TAC.2007.895897","article-title":"On the rendezvous problem for multiple nonholonomic agents","volume":"52","author":"Dimarogonas","year":"2007","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1109\/TMECH.2006.871100","article-title":"Localization and follow-the-leader control of a heterogeneous group of mobile robots","volume":"11","author":"Huang","year":"2006","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1109\/TSMCB.2008.2007810","article-title":"Decentralized robust adaptive control for the multiagent system consensus problem using neural networks","volume":"39","author":"Hou","year":"2009","journal-title":"IEEE Trans. Syst. Man Cybern. Part B (Cybern.)"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.sysconle.2007.07.003","article-title":"Adaptive design for reference velocity recovery in motion coordination","volume":"57","author":"Bai","year":"2008","journal-title":"Syst. Control Lett."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Khan, Q., and Akmeliawati, R. (2017, January 17\u201320). Robust cooperative tracking protocol design for networked higher order nonlinear systems via adaptive second order sliding mode. Proceedings of the 2017 11th Asian Control Conference (ASCC), Gold Coast, Australia.","DOI":"10.1109\/ASCC.2017.8287551"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1109\/TNN.2010.2050601","article-title":"Neural-network-based adaptive leader-following control for multiagent systems with uncertainties","volume":"21","author":"Cheng","year":"2010","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1109\/TMECH.2009.2014057","article-title":"Robust finite-time consensus tracking algorithm for multirobot systems","volume":"14","author":"Khoo","year":"2009","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2014","DOI":"10.1016\/j.automatica.2010.08.008","article-title":"Distributed adaptive control for synchronization of unknown nonlinear networked systems","volume":"46","author":"Das","year":"2010","journal-title":"Automatica"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1002\/rnc.1647","article-title":"Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities","volume":"21","author":"Das","year":"2011","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2273","DOI":"10.1002\/rnc.5946","article-title":"Robust event-triggered consensus tracking control of high-order uncertain nonlinear systems","volume":"32","author":"Yang","year":"2022","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yong, K., Chen, M., and Wu, Q. (2021, January 10\u201312). Finite-Time Performance Recovery Strategy-based NCE Adaptive Neural Control for Networked Nonlinear Systems against DoS Attack. Proceedings of the 2021 4th IEEE International Conference on Industrial Cyber-Physical Systems (ICPS), Victoria, BC, Canada.","DOI":"10.1109\/ICPS49255.2021.9468231"},{"key":"ref_18","first-page":"125749","article-title":"A novel approach for adaptive H\u221e leader-following consensus of higher-order locally Lipschitz multi-agent systems","volume":"395","author":"Basheer","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4105546","DOI":"10.1155\/2022\/4105546","article-title":"Intelligent L2-L\u221e Consensus of Multiagent Systems under Switching Topologies via Fuzzy Deep Q Learning","volume":"2022","author":"Cheng","year":"2022","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1007\/s12555-016-0057-z","article-title":"Neural-network-based integral sliding-mode tracking control of second-order multi-agent systems with unmatched disturbances and completely unknown dynamics","volume":"15","author":"Ma","year":"2017","journal-title":"Int. J. Control. Autom. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.isatra.2018.10.041","article-title":"Robust finite time cooperative control of second order agents: A multi-input multi-output higher order super-twisting based approach","volume":"86","author":"Kamal","year":"2019","journal-title":"ISA Trans."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"147","DOI":"10.24846\/v27i2y201802","article-title":"An integral sliding mode-based robust consensus control protocol design for electro-mechanical systems","volume":"27","author":"Khan","year":"2018","journal-title":"Stud. Inform. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1016\/j.automatica.2014.02.028","article-title":"Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots","volume":"50","author":"Wang","year":"2014","journal-title":"Automatica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.isatra.2021.02.045","article-title":"Neuro-adaptive fast integral terminal sliding mode control design with variable gain robust exact differentiator for under-actuated quadcopter UAV","volume":"120","author":"Ullah","year":"2021","journal-title":"ISA Trans."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ullah, S., Khan, Q., and Mehmood, A. (2022). Neuro-adaptive Fixed-time Non-singular Fast Terminal Sliding Mode Control Design for a Class of Under-actuated Nonlinear Systems. Int. J. Control.","DOI":"10.21203\/rs.3.rs-494194\/v1"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1002\/rnc.1829","article-title":"Distributed consensus control for multi-agent systems using terminal sliding mode and Chebyshev neural networks","volume":"23","author":"Zou","year":"2013","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2214","DOI":"10.1080\/00207721.2014.980367","article-title":"Synchronisation of high-order MIMO nonlinear systems using distributed neuro-adaptive control","volume":"47","author":"Sarand","year":"2016","journal-title":"Int. J. Syst. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.1016\/j.automatica.2012.05.008","article-title":"Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics","volume":"48","author":"Zhang","year":"2012","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1016\/j.automatica.2013.12.033","article-title":"Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems","volume":"50","author":"Qureshi","year":"2014","journal-title":"Automatica"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.automatica.2015.01.021","article-title":"non-singular fixed-time consensus tracking for second-order multi-agent networks","volume":"54","author":"Zuo","year":"2015","journal-title":"Automatica"},{"key":"ref_31","first-page":"1988","article-title":"Arbitrary-order continuous finite-time sliding mode controller for fixed-time convergence","volume":"65","author":"Mishra","year":"2018","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_32","first-page":"199","article-title":"Differential equations with discontinuous right-hand sides","volume":"62","author":"Fillipov","year":"1988","journal-title":"Am. Math. Soc. Trans."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Edwards, C., and Spurgeon, S. (1998). Sliding Mode Control: Theory and Applications, CRC Press.","DOI":"10.1201\/9781498701822"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/7\/2748\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:49:05Z","timestamp":1760136545000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/7\/2748"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,2]]},"references-count":33,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["s22072748"],"URL":"https:\/\/doi.org\/10.3390\/s22072748","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,2]]}}}