{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T03:01:07Z","timestamp":1763348467418,"version":"3.37.3"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T00:00:00Z","timestamp":1660089600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T00:00:00Z","timestamp":1660089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Minufiya University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Process Lett"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper introduces a novel structure of a polynomial weighted output recurrent neural network (PWORNN) for designing an adaptive proportional\u2014integral\u2014derivative (PID) controller. The proposed adaptive PID controller structure based on a polynomial weighted output recurrent neural network (APID-PWORNN) is introduced. In this structure, the number of tunable parameters for the PWORNN only depends on the number of hidden neurons and it is independent of the number of external inputs. The proposed structure of the PWORNN aims to reduce the number of tunable parameters, which reflects on the reduction of the computation time of the proposed algorithm. To guarantee the stability, the optimization, and speed up the convergence of the tunable parameters, i.e., output weights, the proposed network is trained using Lyapunov stability criterion based on an adaptive learning rate. Moreover, by applying the proposed scheme to a nonlinear mathematical system and the heat exchanger system, the robustness of the proposed APID-PWORNN controller has been investigated in this paper and proven its superiority to deal with the nonlinear dynamical systems considering the system parameters uncertainties, disturbances, set-point change, and sensor measurement uncertainty.<\/jats:p>","DOI":"10.1007\/s11063-022-10989-1","type":"journal-article","created":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T12:02:40Z","timestamp":1660132960000},"page":"2885-2910","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Polynomial Recurrent Neural Network-Based Adaptive PID Controller With Stable Learning Algorithm"],"prefix":"10.1007","volume":"55","author":[{"given":"Youssef F.","family":"Hanna","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. Aziz","family":"Khater","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8092-3387","authenticated-orcid":false,"given":"Ahmad M.","family":"El-Nagar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad","family":"El-Bardini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,10]]},"reference":[{"key":"10989_CR1","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.isatra.2017.01.022","volume":"67","author":"R Kumar","year":"2017","unstructured":"Kumar R, Srivastava S, Gupta JRP (2017) Diagonal recurrent neural network based adaptive control of nonlinear dynamical systems using lyapunov stability criterion. ISA Trans 67:407\u2013427","journal-title":"ISA Trans"},{"issue":"9","key":"10989_CR2","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1002\/acs.2916","volume":"32","author":"R Kumar","year":"2018","unstructured":"Kumar R, Srivastava S, Gupta JRP, Mohindru A (2018) Self-recurrent wavelet neural network\u2013based identification and adaptive predictive control of nonlinear dynamical systems. Int J Adapt Control Signal Process 32(9):1326\u20131358","journal-title":"Int J Adapt Control Signal Process"},{"key":"10989_CR3","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1016\/j.isatra.2019.08.032","volume":"98","author":"R Kumar","year":"2020","unstructured":"Kumar R, Srivastava S (2020) Externally Recurrent Neural Network based identification of dynamic systems using Lyapunov stability analysis. ISA Trans 98:292\u2013308","journal-title":"ISA Trans"},{"issue":"3","key":"10989_CR4","doi-asserted-by":"publisher","first-page":"1629","DOI":"10.1007\/s11063-018-9893-6","volume":"49","author":"LA V\u00e1zquez","year":"2019","unstructured":"V\u00e1zquez LA, Jurado F, Casta\u00f1eda CE, Alanis AY (2019) Real-time implementation of a neural integrator backstepping control via recurrent wavelet first order neural network. Neural Process Lett 49(3):1629\u20131648","journal-title":"Neural Process Lett"},{"issue":"1","key":"10989_CR5","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/s11063-019-10081-1","volume":"51","author":"T Liu","year":"2020","unstructured":"Liu T, Liang S, Xiong Q, Wang K (2020) Data-based online optimal temperature tracking control in continuous microwave heating system by adaptive dynamic programming. Neural Process Lett 51(1):167\u2013191","journal-title":"Neural Process Lett"},{"key":"10989_CR6","doi-asserted-by":"crossref","unstructured":"Ma L, Yao Y, Wang M (2016). The optimizing design of wheeled robot tracking system by PID control algorithm based on BP neural network. In: 2016 International Conference on Industrial Informatics-Computing Technology, Intelligent Technology, Industrial Information Integration (ICIICII)\u00a0(pp. 34\u201339). IEEE","DOI":"10.1109\/ICIICII.2016.0020"},{"key":"10989_CR7","doi-asserted-by":"crossref","unstructured":"Li J, G\u00f3mez-Espinosa A (2018) Improving PID Control based on neural network. In: 2018 International Conference on Mechatronics, Electronics and Automotive Engineering (ICMEAE)\u00a0(pp. 186\u2013191). IEEE\u200f","DOI":"10.1109\/ICMEAE.2018.00042"},{"key":"10989_CR8","doi-asserted-by":"crossref","unstructured":"Bari S, Hamdani SSZ, Khan HU, ur Rehman M, Khan H (2019). Artificial neural network based self-tuned PID controller for flight control of quadcopter. In: 2019 International Conference on Engineering and Emerging Technologies (ICEET)\u00a0(pp. 1\u20135). IEEE.","DOI":"10.1109\/CEET1.2019.8711864"},{"key":"10989_CR9","doi-asserted-by":"crossref","first-page":"6988","DOI":"10.1016\/S1876-6102(14)00454-8","volume":"13","author":"L Luoren","year":"2011","unstructured":"Luoren L, Jinling L (2011) Research of PID control algorithm based on neural network. Energy Procedia 13:6988\u20136993","journal-title":"Energy Procedia"},{"key":"10989_CR10","doi-asserted-by":"crossref","unstructured":"Yangxu X, Danhong Z, Huaiun Z, Lianshun W, Yue Q, Zhiwen L (2018) Neural network-fuzzy adaptive PID controller based on VIENNA rectifier. In: Chinese Automation Congress (CAC)\u00a0(pp. 583\u2013588). IEEE\u200f","DOI":"10.1109\/CAC.2018.8623201"},{"key":"10989_CR11","doi-asserted-by":"crossref","unstructured":"Aftab MS, Shafiq M (2015) Adaptive PID controller based on Lyapunov function neural network for time delay temperature control. In: IEEE 8th GCC Conference & Exhibition\u00a0(pp. 1\u20136). IEEE.","DOI":"10.1109\/IEEEGCC.2015.7060094"},{"key":"10989_CR12","doi-asserted-by":"crossref","unstructured":"Jacob R, Murugan S (2016). Implementation of neural network based PID controller. In: 2016 International Conference on electrical, electronics, and optimization techniques (ICEEOT)\u00a0(pp. 2769\u20132771). IEEE.\u200f","DOI":"10.1109\/ICEEOT.2016.7755199"},{"key":"10989_CR13","doi-asserted-by":"crossref","unstructured":"Mahmud K (2013) Neural network based PID control analysis. In: IEEE Global High Tech Congress on Electronics\u00a0(pp. 141\u2013145). Kazemy A, Hosseini SA Farrokhi M (2007). Second order diagonal recurrent neural network. In\u00a02007 IEEE International Symposium on Industrial Electronics\u00a0(pp. 251\u2013256). IEEE","DOI":"10.1109\/ISIE.2007.4374607"},{"key":"10989_CR14","doi-asserted-by":"crossref","unstructured":"Meng Y, Zhiyun Z, Fujian R, Yusong P, Xijie G (2014) Application of adaptive PID based on RBF neural networks in temperature control. In: Proceeding of the 11th World Congress on Intelligent Control and Automation\u00a0(pp. 4302\u20134306). IEEE","DOI":"10.1109\/WCICA.2014.7053436"},{"key":"10989_CR15","doi-asserted-by":"crossref","unstructured":"Kumar R, Srivastava S, Gupta JRP (2016). Artificial neural network based PID controller for online control of dynamical systems. In: IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES)\u00a0(pp. 1\u20136). IEEE.","DOI":"10.1109\/ICPEICES.2016.7853092"},{"issue":"12","key":"10989_CR16","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0243320","volume":"15","author":"J G\u00fcnther","year":"2020","unstructured":"G\u00fcnther J, Reichensd\u00f6rfer E, Pilarski PM, Diepold K (2020) Interpretable PID parameter tuning for control engineering using general dynamic neural networks: an extensive comparison. PLoS ONE 15(12):e0243320","journal-title":"PLoS ONE"},{"key":"10989_CR17","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1007\/978-981-13-1819-1_46","volume-title":"Applications of artificial intelligence techniques in engineering: SIGMA 2018, Volume 1","author":"A Agrawal","year":"2019","unstructured":"Agrawal A, Goyal V, Mishra P (2019) Adaptive control of a nonlinear surge tank-level system using neural network-based PID controller. In: Malik H, Srivastava S, Sood YR, Ahmad A (eds) Applications of artificial intelligence techniques in engineering: SIGMA 2018, Volume 1. Springer, Singapore, pp 491\u2013500. https:\/\/doi.org\/10.1007\/978-981-13-1819-1_46"},{"issue":"1","key":"10989_CR18","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1002\/acs.2955","volume":"33","author":"C Rosales","year":"2019","unstructured":"Rosales C, Soria CM, Rossomando FG (2019) Identification and adaptive PID Control of a hexacopter UAV based on neural networks. Int J Adapt Control Signal Process 33(1):74\u201391","journal-title":"Int J Adapt Control Signal Process"},{"issue":"11","key":"10989_CR19","doi-asserted-by":"publisher","first-page":"1619","DOI":"10.1007\/s12541-018-0189-6","volume":"19","author":"CN Cho","year":"2018","unstructured":"Cho CN, Song YH, Lee CH, Kim HJ (2018) Neural network-based real time PID gain update algorithm for contour error reduction. Int J Precis Eng Manuf 19(11):1619\u20131625","journal-title":"Int J Precis Eng Manuf"},{"issue":"10","key":"10989_CR20","doi-asserted-by":"publisher","first-page":"10656","DOI":"10.1109\/TVT.2020.3019699","volume":"69","author":"Q Pu","year":"2020","unstructured":"Pu Q, Zhu X, Zhang R, Liu J, Cai D, Fu G (2020) Speed profile tracking by an adaptive controller for subway train based on neural network and PID algorithm. IEEE Trans Veh Technol 69(10):10656\u201310667","journal-title":"IEEE Trans Veh Technol"},{"issue":"11","key":"10989_CR21","doi-asserted-by":"publisher","first-page":"11587","DOI":"10.1109\/TIE.2020.3032872","volume":"68","author":"J Hao","year":"2020","unstructured":"Hao J, Zhang G, Liu W, Zheng Y, Ren L (2020) Data-driven tracking control Based on LM and PID neural network with relay feedback for discrete nonlinear systems. IEEE Trans Ind Electron 68(11):11587\u201311597","journal-title":"IEEE Trans Ind Electron"},{"issue":"1","key":"10989_CR22","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1002\/asjc.2356","volume":"23","author":"C Ben Jabeur","year":"2021","unstructured":"Ben Jabeur C, Seddik H (2021) Design of a PID optimized neural networks and PD fuzzy logic controllers for a two-wheeled mobile robot. Asian J Control 23(1):23\u201341","journal-title":"Asian J Control"},{"issue":"1\u20132","key":"10989_CR23","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/0895-7177(95)00225-1","volume":"23","author":"A Patrikar","year":"1996","unstructured":"Patrikar A, Provence J (1996) Nonlinear system identification and adaptive control using polynomial networks. Math Comput Model 23(1\u20132):159\u2013173","journal-title":"Math Comput Model"},{"key":"10989_CR24","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.fss.2018.02.005","volume":"356","author":"T Gonz\u00e1lez","year":"2019","unstructured":"Gonz\u00e1lez T, Sala A, Bernal M (2019) A generalized integral polynomial Lyapunov function for nonlinear systems. Fuzzy Sets Syst 356:77\u201391","journal-title":"Fuzzy Sets Syst"},{"key":"10989_CR25","doi-asserted-by":"crossref","unstructured":"Kazemy A, Hosseini SA, Farrokhi M (2007) Second order diagonal recurrent neural network. In: IEEE International Symposium on Industrial Electronics 2007 (pp. 251-256). IEEE\u200f","DOI":"10.1109\/ISIE.2007.4374607"},{"key":"10989_CR26","doi-asserted-by":"crossref","unstructured":"Lisang L, Xiafu P (2012) Discussion of stability on recurrent neural networks for nonlinear dynamic systems. In: 7th International Conference on Computer Science & Education (ICCSE) (pp. 142-145). IEEE.","DOI":"10.1109\/ICCSE.2012.6295045"},{"issue":"4","key":"10989_CR27","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.isatra.2011.06.005","volume":"50","author":"J Peng","year":"2011","unstructured":"Peng J, Dubay R (2011) Identification and adaptive neural network control of a DC motor system with dead-zone characteristics. ISA Trans 50(4):588\u2013598","journal-title":"ISA Trans"},{"key":"10989_CR28","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1016\/j.neucom.2018.01.073","volume":"287","author":"R Kumar","year":"2018","unstructured":"Kumar R, Srivastava S, Gupta JRP, Mohindru A (2018) Diagonal recurrent neural network based identification of nonlinear dynamical systems with Lyapunov stability based adaptive learning rates. Neurocomputing 287:102\u2013117","journal-title":"Neurocomputing"},{"key":"10989_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.autcon.2021.103722","volume":"127","author":"H Feng","year":"2021","unstructured":"Feng H, Ma W, Yin C, Cao D (2021) Trajectory control of electro-hydraulic position servo system using improved PSO-PID controller. Autom Constr 127:103722","journal-title":"Autom Constr"},{"issue":"12","key":"10989_CR30","doi-asserted-by":"publisher","first-page":"8691","DOI":"10.1007\/s00521-019-04372-w","volume":"32","author":"AA Khater","year":"2020","unstructured":"Khater AA, El-Nagar AM, El-Bardini M, El-Rabaie NM (2020) Online learning based on adaptive learning rate for a class of recurrent fuzzy neural network. Neural Comput Appl 32(12):8691\u20138710","journal-title":"Neural Comput Appl"},{"issue":"4","key":"10989_CR31","first-page":"1037","volume":"11","author":"D Xu","year":"2014","unstructured":"Xu D, Jiang B, Shi P (2014) Adaptive observer based data-driven control for nonlinear discrete-time processes. IEEE Trans Autom Sci Eng 11(4):1037\u20131045","journal-title":"IEEE Trans Autom Sci Eng"},{"issue":"2","key":"10989_CR32","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1002\/aic.690370211","volume":"37","author":"E Eskinat","year":"1991","unstructured":"Eskinat E, Johnson SH, Luyben WL (1991) Use of Hammerstein models in identification of nonlinear systems. AIChE J 37(2):255\u2013268","journal-title":"AIChE J"},{"issue":"11","key":"10989_CR33","doi-asserted-by":"publisher","first-page":"534","DOI":"10.3182\/20130703-3-FR-4038.00129","volume":"46","author":"MA Berger","year":"2013","unstructured":"Berger MA, da Fonseca Neto JV (2013) Neurodynamic programming approach for the PID controller adaptation. IFAC Proc 46(11):534\u2013539","journal-title":"IFAC Proc"},{"key":"10989_CR34","volume-title":"Fractional-order models for nuclear reactor analysis","author":"GE Paredes","year":"2020","unstructured":"Paredes GE (2020) Fractional-order models for nuclear reactor analysis. Woodhead Publishing"}],"container-title":["Neural Processing Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-022-10989-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11063-022-10989-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-022-10989-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T03:47:21Z","timestamp":1727754441000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11063-022-10989-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,10]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["10989"],"URL":"https:\/\/doi.org\/10.1007\/s11063-022-10989-1","relation":{},"ISSN":["1370-4621","1573-773X"],"issn-type":[{"type":"print","value":"1370-4621"},{"type":"electronic","value":"1573-773X"}],"subject":[],"published":{"date-parts":[[2022,8,10]]},"assertion":[{"value":"25 July 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 August 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no conflict of interest between the authors to publish this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}