{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T23:46:00Z","timestamp":1740181560769,"version":"3.37.3"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T00:00:00Z","timestamp":1668988800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T00:00:00Z","timestamp":1668988800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100012687","name":"Universit\u00e4t Kassel","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100012687","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["SN COMPUT. SCI."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This work studies receding-horizon control of discrete-time switched linear systems subject to polytopic constraints for the continuous states and inputs. The objective is to approximate the optimal receding-horizon control strategy for cases in which the online computation is intractable due to the necessity of solving mixed-integer quadratic programs in each discrete time instant. The proposed approach builds upon an approximated optimal finite-horizon control law in closed-loop form with guaranteed constraint satisfaction. The paper derives the properties of recursive feasibility and asymptotic stability for the proposed approach. A numerical example is provided for illustration and evaluation of the approach.<\/jats:p>","DOI":"10.1007\/s42979-022-01442-0","type":"journal-article","created":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T23:05:04Z","timestamp":1669158304000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Receding-Horizon Control of Constrained Switched Systems with Neural Networks as Parametric Function Approximators"],"prefix":"10.1007","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4910-8218","authenticated-orcid":false,"given":"Lukas","family":"Markolf","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Olaf","family":"Stursberg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,21]]},"reference":[{"key":"1442_CR1","doi-asserted-by":"crossref","unstructured":"Tarraf DC. Control of cyber-physical systems. In: Lecture Notes in Control and Information Sciences, vol. 449. Heidelberg: Springer; 2013.","DOI":"10.1007\/978-3-319-01159-2"},{"key":"1442_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807930","volume-title":"Handbook of hybrid systems control: theory, tools, applications","author":"J Lunze","year":"2009","unstructured":"Lunze J, Lamnabhi-Lagarrigue F. Handbook of hybrid systems control: theory, tools, applications. Cambridge: Cambridge University Press; 2009."},{"issue":"3","key":"1442_CR3","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s10626-014-0187-5","volume":"25","author":"F Zhu","year":"2015","unstructured":"Zhu F, Antsaklis PJ. Optimal control of hybrid switched systems: a brief survey. Discrete Event Dyn Syst. 2015;25(3):345\u201364.","journal-title":"Discrete Event Dyn Syst"},{"key":"1442_CR4","volume-title":"Optimal control of switched systems with application to networked embedded control systems","author":"D G\u00f6rges","year":"2012","unstructured":"G\u00f6rges D. Optimal control of switched systems with application to networked embedded control systems. Berlin: Logos Verlag; 2012."},{"issue":"11","key":"1442_CR5","doi-asserted-by":"publisher","first-page":"2669","DOI":"10.1109\/TAC.2009.2031574","volume":"54","author":"W Zhang","year":"2009","unstructured":"Zhang W, Hu J, Abate A. On the value functions of the discrete-time switched lqr problem. IEEE Trans Autom Control. 2009;54(11):2669\u201374.","journal-title":"IEEE Trans Autom Control"},{"issue":"7","key":"1442_CR6","doi-asserted-by":"publisher","first-page":"1815","DOI":"10.1109\/TAC.2011.2178649","volume":"57","author":"W Zhang","year":"2012","unstructured":"Zhang W, Hu J, Abate A. Infinite-horizon switched lqr problems in discrete time: a suboptimal algorithm with performance analysis. IEEE Trans Autom Control. 2012;57(7):1815\u201321.","journal-title":"IEEE Trans Autom Control"},{"key":"1442_CR7","doi-asserted-by":"crossref","unstructured":"Zhang W, Hu J. On optimal quadratic regulation for discrete-time switched linear systems. In: Proceedings of the 11th International Conference on hybrid systems: computation and control, 2008; pp. 584\u201397.","DOI":"10.1007\/978-3-540-78929-1_42"},{"issue":"8","key":"1442_CR8","doi-asserted-by":"publisher","first-page":"1249","DOI":"10.1109\/TAC.2006.878720","volume":"51","author":"B Lincoln","year":"2006","unstructured":"Lincoln B, Rantzer A. Relaxed dynamic programming. IEEE Trans Autom Control. 2006;51(8):1249\u201360.","journal-title":"IEEE Trans Autom Control"},{"issue":"5","key":"1442_CR9","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1049\/ip-cta:20050094","volume":"153","author":"A Rantzer","year":"2006","unstructured":"Rantzer A. Relaxed dynamic programming in switching systems. IEE Proc Control Theory Appl. 2006;153(5):567\u201374.","journal-title":"IEE Proc Control Theory Appl"},{"issue":"1","key":"1442_CR10","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1109\/TAC.2010.2085573","volume":"56","author":"D G\u00f6rges","year":"2011","unstructured":"G\u00f6rges D, Izak M, Liu S. Optimal control and scheduling of switched systems. IEEE Trans Autom Control. 2011;56(1):135\u201340.","journal-title":"IEEE Trans Autom Control"},{"key":"1442_CR11","doi-asserted-by":"crossref","unstructured":"Antunes D, Heemels WPMH. Performance analysis of a class of linear quadratic regulators for switched linear systems. In: Proceedings of the 53rd IEEE Conference on decision and control, 2014; pp. 5475\u201380.","DOI":"10.1109\/CDC.2014.7040245"},{"key":"1442_CR12","doi-asserted-by":"crossref","unstructured":"Chen H, Zheng L, Zhang W. Optimal control inspired q-learning for switched linear systems. In: Proceedings of the 2020 American Control Conference, 2020; pp. 4003\u201310.","DOI":"10.23919\/ACC45564.2020.9147818"},{"issue":"9","key":"1442_CR13","doi-asserted-by":"publisher","first-page":"2100","DOI":"10.1109\/TAC.2008.927799","volume":"53","author":"L Gr\u00fcne","year":"2008","unstructured":"Gr\u00fcne L, Rantzer A. On the infinite horizon performance of receding horizon controllers. IEEE Trans Autom Control. 2008;53(9):2100\u201311.","journal-title":"IEEE Trans Autom Control"},{"issue":"4","key":"1442_CR14","doi-asserted-by":"publisher","first-page":"464","DOI":"10.1016\/j.sysconle.2012.01.011","volume":"61","author":"M Balandat","year":"2012","unstructured":"Balandat M, Zhang W, Abate A. On infinite horizon switched lqr problems with state and control constraints. Syst Control Lett. 2012;61(4):464\u201371.","journal-title":"Syst Control Lett"},{"key":"1442_CR15","doi-asserted-by":"crossref","unstructured":"Liu Z, Stursberg O. Optimizing online control of constrained systems with switched dynamics. In: Proceedings of the 2018 European Control Conference, 2018; pp. 788\u201394.","DOI":"10.23919\/ECC.2018.8550446"},{"key":"1442_CR16","doi-asserted-by":"crossref","unstructured":"Markolf L, Stursberg O. Learning-based optimal control of constrained switched linear systems using neural networks. In: Proceedings of the 18th International Conference on informatics in control, automation and robotics, 2021; pp. 90\u201398.","DOI":"10.5220\/0010581600002994"},{"issue":"4","key":"1442_CR17","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"G Cybenko","year":"1989","unstructured":"Cybenko G. Approximation by superpositions of a sigmoidal function. Math Control Signals Syst. 1989;2(4):303\u201314.","journal-title":"Math Control Signals Syst"},{"issue":"5","key":"1442_CR18","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K Hornik","year":"1989","unstructured":"Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Netw. 1989;2(5):359\u201366.","journal-title":"Neural Netw"},{"issue":"6","key":"1442_CR19","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1016\/S0893-6080(05)80131-5","volume":"6","author":"M Leshno","year":"1993","unstructured":"Leshno M, Lin VY, Pinkus A, Schocken S. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Netw. 1993;6(6):861\u20137.","journal-title":"Neural Netw"},{"issue":"7540","key":"1442_CR20","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1038\/nature14236","volume":"518","author":"V Mnih","year":"2015","unstructured":"Mnih V, Kavukcuoglu K, Silver D, Rusu AA, Veness J, Bellemare MG, Graves A, Riedmiller M, Fidjeland AK, Ostrovski G, Petersen S, Beattie C, Sadik A, Antonoglou I, King H, Kumaran D, Wierstra D, Legg S, Hassabis D. Human-level control through deep reinforcement learning. Nature. 2015;518(7540):529\u201333.","journal-title":"Nature"},{"issue":"7587","key":"1442_CR21","doi-asserted-by":"publisher","first-page":"484","DOI":"10.1038\/nature16961","volume":"529","author":"D Silver","year":"2016","unstructured":"Silver D, Huang A, Maddison CJ, Guez A, Sifre L, van den Driessche G, Schrittwieser J, Antonoglou I, Panneershelvam V, Lanctot M, Dieleman S, Grewe D, Nham J, Kalchbrenner N, Sutskever I, Lillicrap T, Leach M, Kavukcuoglu K, Graepel T, Hassabis D. Mastering the game of Go with deep neural networks and tree search. Nature. 2016;529(7587):484\u20139.","journal-title":"Nature"},{"issue":"7676","key":"1442_CR22","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1038\/nature24270","volume":"550","author":"D Silver","year":"2017","unstructured":"Silver D, Schrittwieser J, Simonyan K, Antonoglou I, Huang A, Guez A, Hubert T, Baker L, Lai M, Bolton A, Chen Y, Lillicrap T, Hui F, Sifre L, van den Driessche G, Graepel T, Hassabis D. Mastering the game of Go without human knowledge. Nature. 2017;550(7676):354\u20139.","journal-title":"Nature"},{"issue":"6419","key":"1442_CR23","doi-asserted-by":"publisher","first-page":"1140","DOI":"10.1126\/science.aar6404","volume":"362","author":"D Silver","year":"2018","unstructured":"Silver D, Hubert T, Schrittwieser J, Antonoglou I, Lai M, Guez A, Lanctot M, Sifre L, Kumaran D, Graepel T, Lillicrap T, Simonyan K, Hassabis D. A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play. Science. 2018;362(6419):1140\u20134.","journal-title":"Science"},{"key":"1442_CR24","volume-title":"Neuro-dynamic programming","author":"DP Bertsekas","year":"1996","unstructured":"Bertsekas DP, Tsitsiklis JN. Neuro-dynamic programming. Belmont: Athena Scientific; 1996."},{"key":"1442_CR25","doi-asserted-by":"crossref","unstructured":"Lendaris GG. A retrospective on adaptive dynamic programming for control. In: Proceedings of the International Joint Conference on neural networks, 2009; pp. 1750\u201357.","DOI":"10.1109\/IJCNN.2009.5178716"},{"issue":"3","key":"1442_CR26","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1109\/MCAS.2009.933854","volume":"9","author":"FL Lewis","year":"2009","unstructured":"Lewis FL, Vrabie D. Reinforcement learning and adaptive dynamic programming for feedback control. IEEE Circ Syst Mag. 2009;9(3):32\u201350.","journal-title":"IEEE Circ Syst Mag"},{"issue":"2","key":"1442_CR27","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1007\/s10957-012-0118-2","volume":"156","author":"M Gaggero","year":"2013","unstructured":"Gaggero M, Gnecco G, Sanguineti M. Dynamic programming and value-function approximation in sequential decision problems: error analysis and numerical results. J Optim Theory Appl. 2013;156(2):380\u2013416.","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"1442_CR28","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10589-013-9614-z","volume":"58","author":"M Gaggero","year":"2014","unstructured":"Gaggero M, Gnecco G, Sanguineti M. Approximate dynamic programming for stochastic N-stage optimization with application to optimal consumption under uncertainty. Comput Optim Appl. 2014;58(1):31\u201385.","journal-title":"Comput Optim Appl"},{"key":"1442_CR29","volume-title":"Reinforcement learning: an introduction","author":"RS Sutton","year":"2018","unstructured":"Sutton RS, Barto A. Reinforcement learning: an introduction. Cambridge: MIT Press; 2018."},{"key":"1442_CR30","volume-title":"Reinforcement learning and optimal control","author":"DP Bertsekas","year":"2019","unstructured":"Bertsekas DP. Reinforcement learning and optimal control. Belmont: Athena Scientific; 2019."},{"key":"1442_CR31","doi-asserted-by":"crossref","unstructured":"Chen S, Saulnier K, Atanasov N, Lee DD, Kumar V, Pappas GJ, Morari M. Approximating explicit model predictive control using constrained neural networks. In: Proceedings of the 2018 Annual American Control Conference, 2018; pp. 1520\u201327.","DOI":"10.23919\/ACC.2018.8431275"},{"issue":"3","key":"1442_CR32","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1109\/LCSYS.2020.2986170","volume":"4","author":"C Cervellera","year":"2020","unstructured":"Cervellera C, Maccio D, Parisini T. Learning robustly stabilizing explicit model predictive controllers: a non-regular sampling approach. IEEE Control Syst Lett. 2020;4(3):737\u201342.","journal-title":"IEEE Control Syst Lett"},{"issue":"9","key":"1442_CR33","doi-asserted-by":"publisher","first-page":"3866","DOI":"10.1109\/TCYB.2020.2999556","volume":"50","author":"B Karg","year":"2020","unstructured":"Karg B, Lucia S. Efficient representation and approximation of model predictive control laws via deep learning. IEEE Trans Cybern. 2020;50(9):3866\u201378.","journal-title":"IEEE Trans Cybern"},{"issue":"3","key":"1442_CR34","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1109\/LCSYS.2020.2980479","volume":"4","author":"JA Paulson","year":"2020","unstructured":"Paulson JA, Mesbah A. Approximate closed-loop robust model predictive control with guaranteed stability and constraint satisfaction. IEEE Control Syst Lett. 2020;4(3):719\u201324.","journal-title":"IEEE Control Syst Lett"},{"key":"1442_CR35","doi-asserted-by":"publisher","DOI":"10.1017\/9781139061759","volume-title":"Predictive control for linear and hybrid systems","author":"F Borrelli","year":"2017","unstructured":"Borrelli F, Bemporad A, Morari M. Predictive control for linear and hybrid systems. Cambridge: Cambridge University Press; 2017."},{"key":"1442_CR36","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"S Boyd","year":"2004","unstructured":"Boyd S, Vandenberghe L. Convex optimization. Cambridge: Cambridge University Press; 2004."},{"key":"1442_CR37","volume-title":"Nonlinear programming","author":"DP Bertsekas","year":"2016","unstructured":"Bertsekas DP. Nonlinear programming. Belmont: Athena Scientific; 2016."},{"key":"1442_CR38","doi-asserted-by":"crossref","unstructured":"Markolf L, Stursberg O. Polytopic input constraints in learning-based optimal control using neural networks. In: Proceedings of the 2021 European Control Conference, 2021; pp. 1018\u201323.","DOI":"10.23919\/ECC54610.2021.9654977"},{"key":"1442_CR39","volume-title":"Deep learning","author":"I Goodfellow","year":"2016","unstructured":"Goodfellow I, Bengio Y, Courville A. Deep learning. Cambridge: MIT Press; 2016."}],"container-title":["SN Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-022-01442-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42979-022-01442-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42979-022-01442-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,7]],"date-time":"2023-01-07T22:26:33Z","timestamp":1673130393000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42979-022-01442-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,21]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["1442"],"URL":"https:\/\/doi.org\/10.1007\/s42979-022-01442-0","relation":{},"ISSN":["2661-8907"],"issn-type":[{"type":"electronic","value":"2661-8907"}],"subject":[],"published":{"date-parts":[[2022,11,21]]},"assertion":[{"value":"17 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}}],"article-number":"62"}}