{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T15:41:47Z","timestamp":1740152507791,"version":"3.37.3"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,6]],"date-time":"2020-12-06T00:00:00Z","timestamp":1607212800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"In the paper we deal with the problem of non-linear dynamic system identification in the presence of random noise. The class of considered systems is relatively general, in the sense that it is not limited to block-oriented structures such as Hammerstein or Wiener models. It is shown that the proposed algorithm can be generalized for two-stage strategy. In step 1 (non-parametric) the system is approximated by multi-dimensional regression functions for a given set of excitations, treated as representative set of points in multi-dimensional space. \u2018Curse of dimensionality problem\u2019 is solved by using specific (quantized or periodic) input sequences. Next, in step 2, non-parametric estimates can be plugged into least squares criterion and support model selection and estimation of system parameters. The proposed strategy allows decomposition of the identification problem, which can be of crucial meaning from the numerical point of view. The \u201cestimation points\u201d in step 1 are selected to ensure good task conditioning in step 2. Moreover, non-parametric procedure plays the role of data compression. We discuss the problem of selection of the scale of non-parametric model, and analyze asymptotic properties of the method. Also, the results of simple simulation are presented, to illustrate functioning of the method. Finally, the proposed method is successfully applied in Differential Scanning Calorimeter (DSC) to analyze aging processes in chalcogenide glasses.<\/jats:p>","DOI":"10.3390\/a13120328","type":"journal-article","created":{"date-parts":[[2020,12,7]],"date-time":"2020-12-07T03:27:12Z","timestamp":1607311632000},"page":"328","source":"Crossref","is-referenced-by-count":4,"title":["Kernel Identification of Non-Linear Systems with General Structure"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1701-5095","authenticated-orcid":false,"given":"Grzegorz","family":"Mzyk","sequence":"first","affiliation":[{"name":"Faculty of Electronics, Wroc\u0142aw University of Science and Technology, 50-372 Wroc\u0142aw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9957-8069","authenticated-orcid":false,"given":"Zygmunt","family":"Hasiewicz","sequence":"additional","affiliation":[{"name":"Faculty of Electronics, Wroc\u0142aw University of Science and Technology, 50-372 Wroc\u0142aw, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6759-2421","authenticated-orcid":false,"given":"Pawe\u0142","family":"Mielcarek","sequence":"additional","affiliation":[{"name":"Faculty of Electronics, Wroc\u0142aw University of Science and Technology, 50-372 Wroc\u0142aw, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1016\/S0165-1684(00)00231-0","article-title":"A bibliography on nonlinear system identification","volume":"81","author":"Giannakis","year":"2001","journal-title":"Signal Process."},{"key":"ref_2","unstructured":"Schetzen, M. (1980). The Volterra and Wiener Theories of Nonlinear Systems, John Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.automatica.2017.06.033","article-title":"Tensor Network alternating linear scheme for MIMO Volterra system identification","volume":"84","author":"Batselier","year":"2017","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.automatica.2017.04.014","article-title":"Regularized non-parametric Volterra kernel estimation","volume":"82","author":"Birpoutsoukis","year":"2017","journal-title":"Automatica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1109\/TAC.1966.1098387","article-title":"An iterative method for the identification of nonlinear systems using the Hammerstein model","volume":"11","author":"Narendra","year":"1966","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Giri, F., and Bai, E.W. (2010). Block-Oriented Nonlinear System Identification, Springer. Lecture Notes in Control and Information Sciences.","DOI":"10.1007\/978-1-84996-513-2"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1929","DOI":"10.1109\/TAC.2004.837592","article-title":"Convergence of the iterative Hammerstein system identification algorithm","volume":"49","author":"Bai","year":"2004","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0005-1098(82)90022-X","article-title":"Identification of systems containing linear dynamic and static nonlinear elements","volume":"18","author":"Billings","year":"1982","journal-title":"Automatica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1109\/TAC.1971.1099787","article-title":"A noniterative method for identification using Hammerstein model","volume":"16","author":"Chang","year":"1971","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1016\/j.automatica.2011.05.008","article-title":"Parameter identification of Hammerstein systems containing backlash operators with arbitrary-shape parametric borders","volume":"47","author":"Giri","year":"2011","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"\u015aliwi\u0144ski, P. (2013). Lecture Notes in Statistics. Nonlinear System Identification by Haar Wavelets, Springer.","DOI":"10.1007\/978-3-642-29396-2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1080\/00207178208922632","article-title":"Instrumental-variable methods for identification of Hammerstein systems","volume":"35","author":"Stoica","year":"1982","journal-title":"Int. J. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1676","DOI":"10.1109\/78.845925","article-title":"Analysis of stochastic gradient tracking of time-varying polynomial Wiener systems","volume":"48","author":"Bershad","year":"2000","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1109\/TAC.2005.864183","article-title":"Recursive identification for Wiener model with discontinuous piece-wise linear function","volume":"51","author":"Chen","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2697","DOI":"10.1016\/j.automatica.2008.02.016","article-title":"Maximum likelihood identification of Wiener models","volume":"44","author":"Hagenblad","year":"2008","journal-title":"Automatica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1080\/00207170310001599122","article-title":"Identification of FIR Wiener systems with unknown, non-invertible, polynomial non-linearities","volume":"76","author":"Lacy","year":"2003","journal-title":"Int. J. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.sysconle.2006.08.001","article-title":"Parameter identification of Wiener systems with multisegment piecewise-linear nonlinearities","volume":"56","year":"2007","journal-title":"Syst. Control Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1109\/9.333765","article-title":"Convergence analysis of recursive identification algorithms based on the nonlinear Wiener model","volume":"39","author":"Wigren","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_19","unstructured":"Pintelon, R., and Schoukens, J. (2004). System Identification: A Frequency Domain Approach, Wiley-IEEE Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Greblicki, W., and Pawlak, M. (2008). Nonparametric System Identification, Cambridge University Press.","DOI":"10.1017\/CBO9780511536687"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gy\u00f6rfi, L., Kohler, M., Krzy\u017cak, A., and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression, Springer.","DOI":"10.1007\/b97848"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W. (1990). Applied Nonparametric Regression, Cambridge University Press.","DOI":"10.1017\/CCOL0521382483"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wand, M., and Jones, H. (1995). Kernel Smoothing, Chapman and Hall.","DOI":"10.1007\/978-1-4899-4493-1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1370","DOI":"10.1109\/TAC.2004.832662","article-title":"Combined parametric-nonparametric identification of Hammerstein systems","volume":"48","author":"Hasiewicz","year":"2004","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1080\/00207170802225930","article-title":"Hammerstein system identification by non-parametric instrumental variables","volume":"82","author":"Hasiewicz","year":"2009","journal-title":"Int. J. Control"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1109\/TCSII.2007.901634","article-title":"A censored sample mean approach to nonparametric identification of nonlinearities in Wiener systems","volume":"54","author":"Mzyk","year":"2007","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mzyk, G. (2014). Combined Parametric-Nonparametric Identification of Block-Oriented Systems, Springer. Lecture Notes in Control and Information Sciences.","DOI":"10.1007\/978-3-319-03596-3"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.automatica.2017.06.038","article-title":"Kernel-based identification of Wiener-Hammerstein system","volume":"83","author":"Mzyk","year":"2017","journal-title":"Automatica"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1002\/acs.3124","article-title":"Wiener system identification by input injection method","volume":"34","author":"Mzyk","year":"2020","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1002\/acs.2584","article-title":"Direct identification of the linear block in Wiener system","volume":"30","author":"Wachel","year":"2016","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kozdra\u015b, B., Mzyk, G., and Mielcarek, P. (2020, January 27\u201329). Identification of the heating process in Differential Scanning Calorimetry with the use of Hammerstein model. Proceedings of the 2020 21st International Carpathian Control Conference (ICCC), High Tatras, Slovakia.","DOI":"10.1109\/ICCC49264.2020.9257252"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1109\/TCS.1985.1085649","article-title":"Fading memory and the problem of approximating nonlinear operators with Volterra series","volume":"32","author":"Boyd","year":"1985","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_33","unstructured":"Van der Vaart, A.W. (2000). Asymptotic Statistics, Cambridge University Press."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/12\/328\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,6]],"date-time":"2024-07-06T06:19:30Z","timestamp":1720246770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/12\/328"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,6]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["a13120328"],"URL":"https:\/\/doi.org\/10.3390\/a13120328","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2020,12,6]]}}}