{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T10:54:34Z","timestamp":1761648874081,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T00:00:00Z","timestamp":1617753600000},"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 scope of this paper is the design of an interval observer bundle for the guaranteed state estimation of an uncertain induction machine with linear, time-varying dynamics. These guarantees are of particular interest in the case of safety-critical systems. In many cases, interval observers provide large intervals for which the usability becomes impractical. Hence, based on a reduced-order hybrid interval observer structure, the guaranteed enclosure within intervals of the magnetizing current\u2019s estimates is improved using a bundle of interval observers. One advantage of such an interval observer bundle is the possibility to reinitialize the interval observers at specified timesteps during runtime with smaller initial intervals, based on previously observed system states, resulting in decreasing interval widths. Thus, unstable observer dynamics are considered so as to take advantage of their transient behavior, whereby the overall stability of the interval estimation is maintained. An algorithm is presented to determine the parametrization of reduced-order interval observers. To this, an adaptive observer gain is introduced with which the system states are observed optimally by considering a minimal interval width at variable operating points. Furthermore, real-time capability and validation of the proposed methods are shown. The results are discussed with simulations as well as experimental data obtained with a test bench.<\/jats:p>","DOI":"10.3390\/s21082584","type":"journal-article","created":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T11:31:59Z","timestamp":1617795119000},"page":"2584","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Guaranteed State Estimation Using a Bundle of Interval Observers with Adaptive Gains Applied to the Induction Machine"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4544-6554","authenticated-orcid":false,"given":"Manuel","family":"Schwartz","sequence":"first","affiliation":[{"name":"Institute of Control Systems, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Krebs","sequence":"additional","affiliation":[{"name":"Institute of Control Systems, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S\u00f6ren","family":"Hohmann","sequence":"additional","affiliation":[{"name":"Institute of Control Systems, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Krebs, S., Pfeifer, M., Fugel, S., Weigold, J., and Hohmann, S. (2016, January 12\u201314). Interval observer for LPV systems based on time-variant transformations. Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, USA.","DOI":"10.1109\/CDC.2016.7798889"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/TASE.2015.2497256","article-title":"Development and Validation of an Automated Steering Control System for Bus Revenue Service","volume":"13","author":"Huang","year":"2016","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1563","DOI":"10.1109\/TASE.2017.2720177","article-title":"A Wavelet-Based Statistical Approach for Monitoring and Diagnosis of Compound Faults with Application to Rolling Bearings","volume":"15","author":"Fan","year":"2017","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1109\/TASE.2013.2264809","article-title":"Fault-Tolerant Control for Safety of Discrete-Event Systems","volume":"11","author":"Shu","year":"2014","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Meskin, N., and Khorasani, K. (2011). Fault Detection and Isolation, Springer.","DOI":"10.1007\/978-1-4419-8393-0"},{"key":"ref_6","first-page":"10124","article-title":"Robust Consistency-Based Diagnosis of Nonlinear Systems by Set Observation","volume":"41","author":"Wolff","year":"2008","journal-title":"IFAC World Congr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2525","DOI":"10.1080\/00207179.2020.1829850","article-title":"Special issue on interval estimation applied to diagnosis and control of uncertain systems","volume":"93","author":"Raissi","year":"2020","journal-title":"Int. J. Control"},{"key":"ref_8","unstructured":"Ding, S.X. (2008). Model-Based Fault Diagnosis Techniques, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sun, L., Alkhatib, H., Kargoll, B., Kreinovich, V., and Neumann, I. (2019). Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems, MDPI.","DOI":"10.3390\/math7121168"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.conengprac.2016.09.004","article-title":"Reduced-order Hybrid Interval Observer for Verified State Estimation of an Induction Machine","volume":"57","author":"Krebs","year":"2016","journal-title":"Control Eng. Pract."},{"key":"ref_11","unstructured":"Durieu, C., Loron, L., Sedda, E., and Zein, I. (September, January 31). Fault Detection of an Induction Motor by Set-Membership Filtering and Kalman Filtering. Proceedings of the European Control Conference, Karlsruhe, Germany."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/B:NUMA.0000049466.96588.a6","article-title":"Guaranteed nonlinear state estimator for cooperative systems","volume":"37","author":"Kieffer","year":"2004","journal-title":"Numer. Alg."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Jaulin, L., Kieffer, M., Didrit, O., and Walter, E. (2001). Applied Interval Analysis, Springer.","DOI":"10.1007\/978-1-4471-0249-6"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zbranek, P., and Vesely, L. (2010, January 21\u201323). Nonlinear state estimation using interval computation in PMSM state observer simulation. Proceedings of the International Conference on Autonomous and Intelligent Systems, Povoa de Varzim, Portugal.","DOI":"10.1109\/AIS.2010.5547030"},{"key":"ref_15","first-page":"2794","article-title":"Interval observers for LPV systems and application to the guaranteed state estimation of an induction machine","volume":"50","author":"Krebs","year":"2017","journal-title":"IFAC World Congr."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1002\/acs.1208","article-title":"Interval observer design based on nonlinear hybridization and practical stability analysis","volume":"25","author":"Meslem","year":"2011","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ramdani, N., Meslem, N., and Candau, Y. (2008). Reachability of Uncertain Nonlinear Systems Using a Nonlinear Hybridization. Hybrid Syst. Comput. Control, 415\u2013428.","DOI":"10.1007\/978-3-540-78929-1_30"},{"key":"ref_18","first-page":"619","article-title":"\u00dcber das Fundamentaltheorem in der Theorie der gew\u00f6hnlichen Differentialgleichungen","volume":"26","year":"1926","journal-title":"Math. Z."},{"key":"ref_19","unstructured":"Smith, H.L. (1995). Monotone Dynamical Systems. Am. Math. Soc., 41."},{"key":"ref_20","first-page":"43","article-title":"Interval observers for non monotone systems application to bioprocess models","volume":"38","author":"Moisan","year":"2005","journal-title":"IFAC World Congr."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1016\/j.jprocont.2003.12.006","article-title":"Closed loop observers bundle for uncertain biotechnological models","volume":"14","author":"Bernard","year":"2004","journal-title":"J. Process Control"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mairet, F., and Bernard, O. (2009, January 15\u201318). Coupling framers to get enhanced interval observers. Application to growth rate estimation in a photobioreactor. Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China.","DOI":"10.1109\/CDC.2009.5400600"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.automatica.2008.07.006","article-title":"Near optimal interval observers bundle for uncertain bioreactors","volume":"45","author":"Moisan","year":"2009","journal-title":"Automatica"},{"key":"ref_24","first-page":"8024","article-title":"Improved Fault Detection and Isolation Strategy using a Bank of Interval Observers","volume":"47","author":"Xu","year":"2014","journal-title":"IFAC World Congr."},{"key":"ref_25","first-page":"hal-02634537","article-title":"Estimation in uncertain switched systems using a bank of interval observers: Local vs glocal approach","volume":"2020","author":"Rotondo","year":"2020","journal-title":"IFAC Wrold Congr."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Krebs, S., Fugel, S., and Hohmann, S. (2017, January 27\u201330). Interval State Observer Based on a Time-Variant Transformation for LPV Systems and Application to Induction Machines. Proceedings of the IEEE Conference on Control Technology and Applications, Maui, HI, USA.","DOI":"10.1109\/CCTA.2017.8062602"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.automatica.2018.06.012","article-title":"Coupled boundary interval observer for LPV systems subject to uncertainties in input, output and parameters","volume":"95","author":"Krebs","year":"2018","journal-title":"Automatica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1684","DOI":"10.1109\/TAC.2003.817920","article-title":"Monotone Control Systems","volume":"48","author":"Angeli","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_29","unstructured":"Cellier, F.E., and Kofman, E. (2006). Continuous System Simulation, Springer."},{"key":"ref_30","unstructured":"Li, Z., Zhang, G., Diao, L., and Liu, Z. (2008, January 3\u20135). Extended Kalman Filter based on inverse gamma model of induction motor. Proceedings of the Vehicle Power and Propulsion Conference, Harbin, China."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Quang, N.P., and Dittrich, J. (2015). Vector Control of Three-Phase AC Machines: System Development in the Practice, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-46915-6"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Karmakar, S., Chattopadhyah, S., Mitra, M., and Sengupta, S. (2016). Induction Motor Fault Diagnosis, Springer.","DOI":"10.1007\/978-981-10-0624-1"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1137\/050638448","article-title":"On Taylor Model Based Integration of ODEs","volume":"45","author":"Neher","year":"2007","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_34","unstructured":"Nocedal, J., and Wright, S. (2006). Numerical Optimization, Springer."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Krebs, S., K\u00f6hrer, L., and Hohmann, S. (2016, January 19\u201321). Interval modelling of a voltage source inverter. Proceedings of the 8th International Conference on Power Electronics, Machines and Drives (PEMD 2016), Glasgow, UK.","DOI":"10.1049\/cp.2016.0336"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/8\/2584\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:34:13Z","timestamp":1760362453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/8\/2584"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,7]]},"references-count":35,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["s21082584"],"URL":"https:\/\/doi.org\/10.3390\/s21082584","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,4,7]]}}}