{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T03:32:42Z","timestamp":1780371162228,"version":"3.54.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T00:00:00Z","timestamp":1717372800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T00:00:00Z","timestamp":1717372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61403056"],"award-info":[{"award-number":["61403056"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61671099"],"award-info":[{"award-number":["61671099"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation Guidance Project of Liaoning Province","award":["2019-ZD-0128"],"award-info":[{"award-number":["2019-ZD-0128"]}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2018YFD0901002"],"award-info":[{"award-number":["2018YFD0901002"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int. J. Fuzzy Syst."],"published-print":{"date-parts":[[2024,11]]},"DOI":"10.1007\/s40815-024-01750-y","type":"journal-article","created":{"date-parts":[[2024,6,3]],"date-time":"2024-06-03T09:01:46Z","timestamp":1717405306000},"page":"2616-2631","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Novel Neuro-fuzzy Learning Algorithm for First-Order Takagi\u2013Sugeno Fuzzy Model: Caputo Fractional-Order Gradient Descent Method"],"prefix":"10.1007","volume":"26","author":[{"given":"Yan","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuanquan","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Shao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rui","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Lv","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,6,3]]},"reference":[{"key":"1750_CR1","doi-asserted-by":"publisher","first-page":"1839","DOI":"10.1007\/s40815-021-01242-3","volume":"24","author":"G Xue","year":"2022","unstructured":"Xue, G., Lin, F., Li, S., Liu, H.: Composite learning control of uncertain fractional-order nonlinear systems with actuator faults based on command filtering and fuzzy approximation. Int. J. Fuzzy Syst. 24, 1839\u20131858 (2022)","journal-title":"Int. J. Fuzzy Syst."},{"key":"1750_CR2","doi-asserted-by":"publisher","first-page":"1046","DOI":"10.1007\/s40815-021-01189-5","volume":"24","author":"M Liang","year":"2022","unstructured":"Liang, M., et al.: Observer-based adaptive fuzzy output feedback control for a class of fractional-order nonlinear systems with full-state constraints. Int. J. Fuzzy Syst. 24, 1046\u20131058 (2022)","journal-title":"Int. J. Fuzzy Syst."},{"key":"1750_CR3","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1007\/s40815-018-0559-3","volume":"21","author":"Y Zhou","year":"2019","unstructured":"Zhou, Y., Wang, H., Liu, H.: Generalized function projective synchronization of incommensurate fractional-order chaotic systems with inputs saturation. Int. J. Fuzzy Syst. 21, 823\u2013836 (2019)","journal-title":"Int. J. Fuzzy Syst."},{"key":"1750_CR4","doi-asserted-by":"crossref","unstructured":"Shi, J., Cao, J., Liu, H., Zhang, X.: Compound adaptive fuzzy output feedback control for uncertain fractional-order nonlinear systems with fuzzy dead-zone input. Int. J. Fuzzy Syst, 1\u201314 (2023)","DOI":"10.1007\/s40815-022-01457-y"},{"key":"1750_CR5","doi-asserted-by":"publisher","first-page":"102599","DOI":"10.1109\/ACCESS.2022.3203067","volume":"10","author":"A Daoui","year":"2022","unstructured":"Daoui, A., et al.: Biomedical multimedia encryption by fractional-order Meixner polynomials map and quaternion fractional-order Meixner moments. IEEE Access 10, 102599\u2013102617 (2022)","journal-title":"IEEE Access"},{"key":"1750_CR6","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1109\/TNNLS.2013.2286175","volume":"26","author":"Y-F Pu","year":"2013","unstructured":"Pu, Y.-F., et al.: Fractional extreme value adaptive training method: fractional steepest descent approach. IEEE Trans. Neural Netw. Learn. Syst. 26, 653\u2013662 (2013)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"1750_CR7","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.neunet.2017.02.007","volume":"89","author":"J Wang","year":"2017","unstructured":"Wang, J., Wen, Y., Gou, Y., Ye, Z., Chen, H.: Fractional-order gradient descent learning of bp neural networks with Caputo derivative. Neural Netw. 89, 19\u201330 (2017)","journal-title":"Neural Netw."},{"key":"1750_CR8","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.neucom.2019.10.017","volume":"408","author":"D Sheng","year":"2020","unstructured":"Sheng, D., Wei, Y., Chen, Y., Wang, Y.: Convolutional neural networks with fractional order gradient method. Neurocomputing 408, 42\u201350 (2020)","journal-title":"Neurocomputing"},{"key":"1750_CR9","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/j.asoc.2019.02.023","volume":"78","author":"E Yazid","year":"2019","unstructured":"Yazid, E., Garratt, M., Santoso, F.: Position control of a quadcopter drone using evolutionary algorithms-based self-tuning for first-order Takagi-Sugeno-Kang fuzzy logic autopilots. Appl. Soft Comput. 78, 373\u2013392 (2019)","journal-title":"Appl. Soft Comput."},{"key":"1750_CR10","doi-asserted-by":"publisher","first-page":"1258","DOI":"10.1093\/ietele\/e90-c.6.1258","volume":"90","author":"M Mottaghi-Kashtiban","year":"2007","unstructured":"Mottaghi-Kashtiban, M., Khoei, A., Hadidi, K.: A current-mode, first-order Takagi-Sugeno-Kang fuzzy logic controller, supporting rational-powered membership functions. IEICE Trans. Electron. 90, 1258\u20131266 (2007)","journal-title":"IEICE Trans. Electron."},{"key":"1750_CR11","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1109\/TSMC.1985.6313399","volume":"SMC\u201315","author":"T Takagi","year":"1985","unstructured":"Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. B SMC\u201315, 116\u2013132 (1985)","journal-title":"IEEE Trans. Syst. Man Cybern. B"},{"key":"1750_CR12","doi-asserted-by":"publisher","first-page":"990","DOI":"10.1109\/21.384264","volume":"25","author":"Y Jin","year":"1995","unstructured":"Jin, Y., Jiang, J., Zhu, J.: Neural network based fuzzy identification and its application to modeling and control of complex systems. IEEE Trans. Syst. Man Cybern. B 25, 990\u2013997 (1995)","journal-title":"IEEE Trans. Syst. Man Cybern. B"},{"key":"1750_CR13","unstructured":"Liu, Y., Yang, J., Yang, D., Wu, W.: A modified gradient-based neuro-fuzzy learning algorithm for pi-sigma network based on first-order Takagi-Sugeno system. J. Math. Res. Appl. 34 (2014)"},{"key":"1750_CR14","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1112\/jlms\/s2-3.2.241","volume":"2","author":"ER Love","year":"1971","unstructured":"Love, E.R.: Fractional derivatives of imaginary order. J. Lond. Math. Soc. 2, 241\u2013259 (1971)","journal-title":"J. Lond. Math. Soc."},{"key":"1750_CR15","volume-title":"The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order","author":"K Oldham","year":"1974","unstructured":"Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974)"},{"key":"1750_CR16","doi-asserted-by":"publisher","first-page":"1322","DOI":"10.1016\/j.camwa.2007.11.052","volume":"56","author":"D Liu","year":"2008","unstructured":"Liu, D., Naadimuthu, G., Lee, E.: Trajectory tracking in aircraft landing operations management using the adaptive neural fuzzy inference system. Comput. Math. Appl. 56, 1322\u20131327 (2008)","journal-title":"Comput. Math. Appl."},{"key":"1750_CR17","doi-asserted-by":"publisher","first-page":"1428","DOI":"10.1016\/j.asoc.2007.10.010","volume":"8","author":"KT Chaturvedi","year":"2008","unstructured":"Chaturvedi, K.T., Pandit, M., Srivastava, L.: Modified neo-fuzzy neuron-based approach for economic and environmental optimal power dispatch. Appl. Soft Comput. 8, 1428\u20131438 (2008)","journal-title":"Appl. Soft Comput."},{"key":"1750_CR18","doi-asserted-by":"publisher","first-page":"1630","DOI":"10.1016\/j.ins.2009.12.030","volume":"180","author":"W Wu","year":"2010","unstructured":"Wu, W., Li, L., Yang, J., Liu, Y.: A modified gradient-based neuro-fuzzy learning algorithm and its convergence. Inf. Sci. 180, 1630\u20131642 (2010)","journal-title":"Inf. Sci."},{"key":"1750_CR19","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/j.neucom.2014.01.041","volume":"138","author":"Y Liu","year":"2014","unstructured":"Liu, Y., Wu, W., Fan, Q., Yang, D., Wang, J.: A modified gradient learning algorithm with smoothing l1\/2 regularization for Takagi-Sugeno fuzzy models. Neurocomputing 138, 229\u2013237 (2014)","journal-title":"Neurocomputing"},{"key":"1750_CR20","doi-asserted-by":"publisher","first-page":"1064","DOI":"10.1109\/TNN.2002.1031939","volume":"13","author":"N Ampazis","year":"2002","unstructured":"Ampazis, N., Perantonis, S.J.: Two highly efficient second-order algorithms for training feedforward networks. IEEE Trans. Neural Netw. 13, 1064\u20131074 (2002)","journal-title":"IEEE Trans. Neural Netw."},{"key":"1750_CR21","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/0888-613X(93)90011-2","volume":"9","author":"H Ichihashi","year":"1993","unstructured":"Ichihashi, H., T\u00fcrksen, I.B.: A neuro-fuzzy approach to data analysis of pairwise comparisons. Int. J. Approx. Reason. 9, 227\u2013248 (1993)","journal-title":"Int. J. Approx. Reason."}],"container-title":["International Journal of Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40815-024-01750-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40815-024-01750-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40815-024-01750-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T14:08:21Z","timestamp":1737641301000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40815-024-01750-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,3]]},"references-count":21,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["1750"],"URL":"https:\/\/doi.org\/10.1007\/s40815-024-01750-y","relation":{},"ISSN":["1562-2479","2199-3211"],"issn-type":[{"value":"1562-2479","type":"print"},{"value":"2199-3211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,3]]},"assertion":[{"value":"6 September 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 March 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}