{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T21:02:41Z","timestamp":1758142961792,"version":"3.44.0"},"reference-count":55,"publisher":"Walter de Gruyter GmbH","issue":"8","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,8,26]]},"abstract":"<jats:title>Zusammenfassung<\/jats:title>\n               <jats:p>Dieser Beitrag gibt eine einf\u00fchrende \u00dcbersicht zur Beschreibung konzentriert- und verteilt-parametrischer Systeme im Observablenraum mittels des Koopman-Operators. Diese Darstellung wird anschlie\u00dfend genutzt, um f\u00fcr unbekannte lineare parabolische Systeme datenbasiert Regler zur robusten asymptotischen St\u00f6rkompensation zu entwerfen. Dabei werden zeitdiskrete Randmessungen und ein unbekanntes St\u00f6rmodell angenommen. Die L\u00f6sung dieser Problemstellung erfolgt auf Grundlage eines neuen Koopman-basierten Parameteridentifikationsverfahrens, welches einen Reglerentwurf ohne Systemapproximation mittels der Backstepping-Methodik erm\u00f6glicht. Ein parabolisches Beispielsystem veranschaulicht die Ergebnisse des Beitrags.<\/jats:p>","DOI":"10.1515\/auto-2025-0037","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T14:00:25Z","timestamp":1754488825000},"page":"622-635","source":"Crossref","is-referenced-by-count":0,"title":["Datenbasierte robuste St\u00f6rkompensation f\u00fcr lineare parabolische Systeme mittels Koopman-Operatoren"],"prefix":"10.1515","volume":"73","author":[{"given":"Joachim","family":"Deutscher","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mess-, Regel- und Mikrotechnik, Universit\u00e4t Ulm , Albert-Einstein-Allee 41, 89081 Ulm , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Julian","family":"Zimmer","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mess-, Regel- und Mikrotechnik, Universit\u00e4t Ulm , Albert-Einstein-Allee 41, 89081 Ulm , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"2025091715210250715_j_auto-2025-0037_ref_001","doi-asserted-by":"crossref","unstructured":"T. Martin, T. Sch\u00f6n, und F. Allg\u00f6wer, \u201cGuarantees for data-driven control of nonlinear systems using semidefinite programming: a survey,\u201d Ann. Rev. Control, vol.\u00a056, 2023, Art. no. 100911. https:\/\/doi.org\/10.1016\/j.arcontrol.2023.100911.","DOI":"10.1016\/j.arcontrol.2023.100911"},{"key":"2025091715210250715_j_auto-2025-0037_ref_002","doi-asserted-by":"crossref","unstructured":"Z.-S. Hou and Z. Wang, \u201cFrom model-based control to data-driven control: survey, classification and perspective,\u201d Inf. Sci., vol.\u00a0235, pp.\u00a03\u201335, 2013. https:\/\/doi.org\/10.1016\/j.ins.2012.07.014.","DOI":"10.1016\/j.ins.2012.07.014"},{"key":"2025091715210250715_j_auto-2025-0037_ref_003","doi-asserted-by":"crossref","unstructured":"S. Brunton and N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge, Cambridge University Press, 2022.","DOI":"10.1017\/9781009089517"},{"key":"2025091715210250715_j_auto-2025-0037_ref_004","doi-asserted-by":"crossref","unstructured":"L. Ecker and M. Sch\u00f6berl, \u201cDatenbasierter Regler- und Beobachterentwurf mit neuronalen kanonischen Strukturen,\u201d at \u2013 Automatisierungstechnik, vol.\u00a072, pp.\u00a0733\u2013744, 2024. https:\/\/doi.org\/10.1515\/auto-2024-0049.","DOI":"10.1515\/auto-2024-0049"},{"key":"2025091715210250715_j_auto-2025-0037_ref_005","doi-asserted-by":"crossref","unstructured":"J. Fiedler, D. Gerbet, and K. R\u00f6benack, \u201cDatenbasierter Entwurf von Einbettungsbeobachtern unter Nutzung von Automatischem Differenzieren,\u201d at \u2013 Automatisierungstechnik, vol.\u00a072, pp.\u00a0745\u2013756, 2024. https:\/\/doi.org\/10.1515\/auto-2024-5066.","DOI":"10.1515\/auto-2024-5066"},{"key":"2025091715210250715_j_auto-2025-0037_ref_006","doi-asserted-by":"crossref","unstructured":"A. Mauroy, I. Mezi\u0107, and Y. Susuki, The Koopman Operator in Systems and Control, London, Cham, Springer, 2020.","DOI":"10.1007\/978-3-030-35713-9"},{"key":"2025091715210250715_j_auto-2025-0037_ref_007","doi-asserted-by":"crossref","unstructured":"R. Curtain and H. Zwart, Introduction to Infinite-Dimensional Systems Theory, New York, Springer-Verlag, 2020.","DOI":"10.1007\/978-1-0716-0590-5"},{"key":"2025091715210250715_j_auto-2025-0037_ref_008","doi-asserted-by":"crossref","unstructured":"M. Williams, I. Kevrekidis, and C. Rowley, \u201cA data-driven approximation of the Koopman operator: extending dynamic mode decomposition,\u201d J. Nonlinear Sci., vol.\u00a025, pp.\u00a01307\u20131346, 2015. https:\/\/doi.org\/10.1007\/s00332-015-9258-5.","DOI":"10.1007\/s00332-015-9258-5"},{"key":"2025091715210250715_j_auto-2025-0037_ref_009","doi-asserted-by":"crossref","unstructured":"M. Korda and I. Mezi\u0107, \u201cOn convergence of extended dynamic mode decomposition to the Koopman operator,\u201d J. Nonlinear Sci., vol.\u00a028, pp.\u00a0687\u2013710, 2018. https:\/\/doi.org\/10.1007\/s00332-017-9423-0.","DOI":"10.1007\/s00332-017-9423-0"},{"key":"2025091715210250715_j_auto-2025-0037_ref_010","doi-asserted-by":"crossref","unstructured":"F. N\u00fcske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, \u201cFinite-data error bounds for Koopman-based prediction and control,\u201d J. Nonlinear Sci., vol.\u00a033, 2023, Art. no. 14. https:\/\/doi.org\/10.1007\/s00332-022-09862-1.","DOI":"10.1007\/s00332-022-09862-1"},{"key":"2025091715210250715_j_auto-2025-0037_ref_011","doi-asserted-by":"crossref","unstructured":"M. Schaller, K. Worthmann, F. Philipp, S. Peitz, and F. N\u00fcske, \u201cTowards reliable data-based optimal and predictive control using extended DMD,\u201d IFAC-PapersOnLine, vol.\u00a056, pp.\u00a0169\u2013174, 2023. https:\/\/doi.org\/10.1016\/j.ifacol.2023.02.029.","DOI":"10.1016\/j.ifacol.2023.02.029"},{"key":"2025091715210250715_j_auto-2025-0037_ref_012","doi-asserted-by":"crossref","unstructured":"L. Bold, L. Gr\u00fcne, M. Schaller, and K. Worthmann, \u201cData-driven MPC with stability guarantees using extended dynamic mode decomposition,\u201d IEEE Trans. Autom. Control, vol.\u00a070, pp.\u00a0534\u2013541, 2025.","DOI":"10.1109\/TAC.2024.3431169"},{"key":"2025091715210250715_j_auto-2025-0037_ref_013","doi-asserted-by":"crossref","unstructured":"R. Str\u00e4sser, M. Schaller, K. Worthmann, J. Berberich, and F. Allg\u00f6wer, \u201cKoopman-based feedback design with stability guarantees,\u201d IEEE Trans. Autom. Control, vol.\u00a070, pp.\u00a0355\u2013370, 2025. https:\/\/doi.org\/10.1109\/tac.2024.3425770.","DOI":"10.1109\/TAC.2024.3425770"},{"key":"2025091715210250715_j_auto-2025-0037_ref_014","doi-asserted-by":"crossref","unstructured":"H. Arbabi, M. Korda, and I. Mezi\u0107, \u201cA data-driven Koopman model predictive control framework for nonlinear partial differential equations,\u201d in IEEE Conf. Dec. Control (CDC), 2018, pp.\u00a06409\u20136414.","DOI":"10.1109\/CDC.2018.8619720"},{"key":"2025091715210250715_j_auto-2025-0037_ref_015","doi-asserted-by":"crossref","unstructured":"S. Peitz and S. Klus, \u201cKoopman operator-based model reduction for switched-system control of PDEs,\u201d Automatica, vol.\u00a0106, pp.\u00a0184\u2013191, 2019. https:\/\/doi.org\/10.1016\/j.automatica.2019.05.016.","DOI":"10.1016\/j.automatica.2019.05.016"},{"key":"2025091715210250715_j_auto-2025-0037_ref_016","doi-asserted-by":"crossref","unstructured":"H. Nakao and I. Mezi\u0107, \u201cSpectral analysis of the Koopman operator for partial differential equations,\u201d Chaos, vol.\u00a030, 2020, Art. no. 113131. https:\/\/doi.org\/10.1063\/5.0011470.","DOI":"10.1063\/5.0011470"},{"key":"2025091715210250715_j_auto-2025-0037_ref_017","doi-asserted-by":"crossref","unstructured":"A. Mauroy, \u201cKoopman operator framework for spectral analysis and identification of infinite-dimensional systems,\u201d Mathematics, vol.\u00a09, 2021, Art. no. 2495. https:\/\/doi.org\/10.3390\/math9192495.","DOI":"10.3390\/math9192495"},{"key":"2025091715210250715_j_auto-2025-0037_ref_018","doi-asserted-by":"crossref","unstructured":"J. Deutscher, \u201cData-driven control of linear parabolic systems using Koopman eigenstructure assignment,\u201d IEEE Trans. Autom. Control, vol.\u00a070, pp.\u00a0665\u2013672, 2025. https:\/\/doi.org\/10.1109\/tac.2024.3441672.","DOI":"10.1109\/TAC.2024.3441672"},{"key":"2025091715210250715_j_auto-2025-0037_ref_019","doi-asserted-by":"crossref","unstructured":"P. Bevanda, S. Sosnowski, and S. Hirche, \u201cKoopman operator dynamical models: learning, analysis and control,\u201d Ann. Rev. Control, vol.\u00a052, pp.\u00a0197\u2013212, 2021. https:\/\/doi.org\/10.1016\/j.arcontrol.2021.09.002.","DOI":"10.1016\/j.arcontrol.2021.09.002"},{"key":"2025091715210250715_j_auto-2025-0037_ref_020","doi-asserted-by":"crossref","unstructured":"S. Brunton, M. Budi\u0161i\u0107, E. Kaiser, and N. Kutz, \u201cModern Koopman theory for dynamical systems,\u201d SIAM Rev., vol.\u00a064, pp.\u00a0229\u2013340, 2022. https:\/\/doi.org\/10.1137\/21m1401243.","DOI":"10.1137\/21M1401243"},{"key":"2025091715210250715_j_auto-2025-0037_ref_021","doi-asserted-by":"crossref","unstructured":"S. Otto and C. Rowley, \u201cKoopman operators for estimation and control of dynamical systems,\u201d Annu. Rev. Control Robot. Auton. Syst., vol.\u00a04, pp.\u00a059\u201387, 2021. https:\/\/doi.org\/10.1146\/annurev-control-071020-010108.","DOI":"10.1146\/annurev-control-071020-010108"},{"key":"2025091715210250715_j_auto-2025-0037_ref_022","doi-asserted-by":"crossref","unstructured":"L. Zhu and Z. Chen, \u201cData informativity for robust output regulation,\u201d IEEE Trans. Autom. Control, vol.\u00a069, pp.\u00a07075\u20137080, 2024. https:\/\/doi.org\/10.1109\/tac.2024.3389742.","DOI":"10.1109\/TAC.2024.3389742"},{"key":"2025091715210250715_j_auto-2025-0037_ref_023","doi-asserted-by":"crossref","unstructured":"H. van Waarde, J. Eising, H. Trentelman, and M. Camlibel, \u201cData informativity: a new perspective on data-driven analysis and control,\u201d IEEE Trans. Autom. Control, vol.\u00a065, pp.\u00a04753\u20134768, 2020. https:\/\/doi.org\/10.1109\/tac.2020.2966717.","DOI":"10.1109\/TAC.2020.2966717"},{"key":"2025091715210250715_j_auto-2025-0037_ref_024","doi-asserted-by":"crossref","unstructured":"J. Deutscher, \u201cData-driven robust output regulation of continuous-time LTI systems using Koopman operators,\u201d IEEE Trans. Autom. Control, vol.\u00a069, pp.\u00a08774\u20138781, 2024. https:\/\/doi.org\/10.1109\/tac.2024.3414708.","DOI":"10.1109\/TAC.2024.3414708"},{"key":"2025091715210250715_j_auto-2025-0037_ref_025","doi-asserted-by":"crossref","unstructured":"B. Kieboom, M. Bartzioka, and M. Jafarian, \u201cUtility of the Koopman operator in output regulation of disturbed nonlinear systems,\u201d in 2023 62nd IEEE Conference on Decision and Control (CDC), 2023, pp.\u00a02616\u20132621.","DOI":"10.1109\/CDC49753.2023.10383637"},{"key":"2025091715210250715_j_auto-2025-0037_ref_026","doi-asserted-by":"crossref","unstructured":"H. Arbabi and I. Mezi\u0107, \u201cErgodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator,\u201d SIAM J. Appl. Dyn. Syst., vol.\u00a016, pp.\u00a02096\u20132126, 2017. https:\/\/doi.org\/10.1137\/17m1125236.","DOI":"10.1137\/17M1125236"},{"key":"2025091715210250715_j_auto-2025-0037_ref_027","doi-asserted-by":"crossref","unstructured":"Z. Drma\u010d, I. Mezi\u0107, and R. Mohr, \u201cData driven modal decompositions: analysis and enhancements,\u201d SIAM J. Sci. Comput., vol.\u00a040, pp.\u00a0A2253\u2013A2285, 2018. https:\/\/doi.org\/10.1137\/17m1144155.","DOI":"10.1137\/17M1144155"},{"key":"2025091715210250715_j_auto-2025-0037_ref_028","doi-asserted-by":"crossref","unstructured":"I. Mezi\u0107, \u201cOn numerical approximations of the Koopman operator,\u201d Mathematics, vol.\u00a010, 2022, Art. no. 1180. https:\/\/doi.org\/10.3390\/math10071180.","DOI":"10.3390\/math10071180"},{"key":"2025091715210250715_j_auto-2025-0037_ref_029","doi-asserted-by":"crossref","unstructured":"V. Kravchenko, Direct and Inverse Sturm-Liouville Problems, Cham, Birk\u00e4user, 2020.","DOI":"10.1007\/978-3-030-47849-0"},{"key":"2025091715210250715_j_auto-2025-0037_ref_030","doi-asserted-by":"crossref","unstructured":"C. Kubrusly, \u201cDistributed parameter system indentification a survey,\u201d Int. J. Control, vol.\u00a026, pp.\u00a0509\u2013535, 1977. https:\/\/doi.org\/10.1080\/00207177708922326.","DOI":"10.1080\/00207177708922326"},{"key":"2025091715210250715_j_auto-2025-0037_ref_031","doi-asserted-by":"crossref","unstructured":"M. Polis, \u201cThe distributed system parameter identification problem: a survey of recent results,\u201d IFAC Proc., vol.\u00a016, pp.\u00a045\u201358, 1983. https:\/\/doi.org\/10.1016\/b978-0-08-029361-5.50011-2.","DOI":"10.1016\/S1474-6670(17)62252-3"},{"key":"2025091715210250715_j_auto-2025-0037_ref_032","doi-asserted-by":"crossref","unstructured":"D. Gay and W. Ray, \u201cIdentification and control of distributed parameter systems by means of the singular value decomposition,\u201d Chem. Eng. Sci., vol.\u00a050, pp.\u00a01519\u20131539, 1995. https:\/\/doi.org\/10.1016\/0009-2509(95)00017-y.","DOI":"10.1016\/0009-2509(95)00017-Y"},{"key":"2025091715210250715_j_auto-2025-0037_ref_033","doi-asserted-by":"crossref","unstructured":"D. Coca and S. Billings, \u201cIdentification of finite dimensional models of infinite dimensional dynamical systems,\u201d Automatica, vol.\u00a038, pp.\u00a01851\u20131865, 2002. https:\/\/doi.org\/10.1016\/s0005-1098(02)00099-7.","DOI":"10.1016\/S0005-1098(02)00099-7"},{"key":"2025091715210250715_j_auto-2025-0037_ref_034","doi-asserted-by":"crossref","unstructured":"T. Kn\u00fcppel and F. Woittennek, \u201cEin Beitrag zur Parameteridentifikation in \u00f6rtlich verteilten Systemen,\u201d at \u2013 Automatisierungstechnik, vol.\u00a061, pp.\u00a0521\u2013532, 2013. https:\/\/doi.org\/10.1524\/auto.2013.1014.","DOI":"10.1524\/auto.2013.1014"},{"key":"2025091715210250715_j_auto-2025-0037_ref_035","doi-asserted-by":"crossref","unstructured":"O. Krivorotko, S. Kabanikhin, S. Zhang, and V. Kashtanova, \u201cGlobal and local optimization in identification of parabolic systems,\u201d in Ill-Posed Prob., vol.\u00a028, J. Inv, Ed., 2020, pp.\u00a0899\u2013913.","DOI":"10.1515\/jiip-2020-0083"},{"key":"2025091715210250715_j_auto-2025-0037_ref_036","doi-asserted-by":"crossref","unstructured":"J. Rudolph and F. Woittennek, \u201cEin algebraischer Zugang zur Parameteridentifikation in linearen unendlichdimensionalen Systemen,\u201d at \u2013 Automatisierungstechnik, vol.\u00a055, pp.\u00a0457\u2013467, 2007. https:\/\/doi.org\/10.1524\/auto.2007.55.9.457.","DOI":"10.1524\/auto.2007.55.9.457"},{"key":"2025091715210250715_j_auto-2025-0037_ref_037","doi-asserted-by":"crossref","unstructured":"N. Gehring and J. Rudolph, \u201cAn algebraic algorithm for parameter identification in a class of systems described by linear partial differential equations,\u201d PAMM, vol.\u00a016, pp.\u00a039\u201342, 2016. https:\/\/doi.org\/10.1002\/pamm.201610011.","DOI":"10.1002\/pamm.201610011"},{"key":"2025091715210250715_j_auto-2025-0037_ref_038","doi-asserted-by":"crossref","unstructured":"A. Mauroy and J. Gonclaves, \u201cParameter estimation and identification for nonlinear systems with the Koopman operator,\u201d in The Koopman Operator in Systems and Control, A. Mauroy, I. Mezi\u0107, and Y. Susuki, Eds., Cham, Springer, 2020, pp.\u00a0335\u2013357.","DOI":"10.1007\/978-3-030-35713-9_13"},{"key":"2025091715210250715_j_auto-2025-0037_ref_039","doi-asserted-by":"crossref","unstructured":"M. Krstic and A. Smyshlyaev, Boundary Control of PDEs \u2014 A Course on Backstepping Designs, Philadelphia, SIAM, 2008.","DOI":"10.1137\/1.9780898718607"},{"key":"2025091715210250715_j_auto-2025-0037_ref_040","doi-asserted-by":"crossref","unstructured":"J. Deutscher, \u201cBackstepping design of robust output feedback regulators for boundary controlled parabolic PDEs,\u201d IEEE Trans. Autom. Control, vol.\u00a061, pp.\u00a02288\u20132294, 2016. https:\/\/doi.org\/10.1109\/tac.2015.2491718.","DOI":"10.1109\/TAC.2015.2491718"},{"key":"2025091715210250715_j_auto-2025-0037_ref_041","doi-asserted-by":"crossref","unstructured":"J. Deutscher and S. Kerschbaum, \u201cRobust output regulation by state feedback control for coupled linear parabolic PIDEs,\u201d IEEE Trans. Autom. Control, vol.\u00a065, pp.\u00a02207\u20132214, 2020. https:\/\/doi.org\/10.1109\/tac.2019.2938329.","DOI":"10.1109\/TAC.2019.2938329"},{"key":"2025091715210250715_j_auto-2025-0037_ref_042","doi-asserted-by":"crossref","unstructured":"J. Deutscher and J. Zimmer, \u201cA Koopman-backstepping approach to data-driven robust output regulation for linear parabolic systems,\u201d in Automatica, 2025. provisorisch angenommen.","DOI":"10.1016\/j.automatica.2025.112538"},{"key":"2025091715210250715_j_auto-2025-0037_ref_043","doi-asserted-by":"crossref","unstructured":"B. O. Koopman, \u201cHamiltonian systems and transformation in Hilbert space,\u201d Proc. Natl. Acad. Sci. U. S. A., vol.\u00a017, pp.\u00a0315\u2013318, 1931. https:\/\/doi.org\/10.1073\/pnas.17.5.315.","DOI":"10.1073\/pnas.17.5.315"},{"key":"2025091715210250715_j_auto-2025-0037_ref_044","doi-asserted-by":"crossref","unstructured":"I. Mezi\u0107, \u201cSpectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,\u201d J. Nonlinear Sci., vol.\u00a030, pp.\u00a02091\u20132145, 2020. https:\/\/doi.org\/10.1007\/s00332-019-09598-5.","DOI":"10.1007\/s00332-019-09598-5"},{"key":"2025091715210250715_j_auto-2025-0037_ref_045","doi-asserted-by":"crossref","unstructured":"J. Deutscher, Zustandsregelung verteilt-parametrischer Systeme, Heidelberg, Springer, 2012.","DOI":"10.1007\/978-3-642-19559-4"},{"key":"2025091715210250715_j_auto-2025-0037_ref_046","doi-asserted-by":"crossref","unstructured":"A. Smyshlyaev and M. Krstic, \u201cBackstepping observers for a class of parabolic PDEs,\u201d Syst. Control Lett., vol.\u00a054, pp.\u00a0613\u2013625, 2005. https:\/\/doi.org\/10.1016\/j.sysconle.2004.11.001.","DOI":"10.1016\/j.sysconle.2004.11.001"},{"key":"2025091715210250715_j_auto-2025-0037_ref_047","doi-asserted-by":"crossref","unstructured":"J. Proctor, S. Brunton, and J. Kutz, \u201cGeneralizing Koopman theory to allow for inputs and control,\u201d SIAM J. Appl. Dyn. Syst., vol.\u00a017, pp.\u00a0909\u2013930, 2018. https:\/\/doi.org\/10.1137\/16m1062296.","DOI":"10.1137\/16M1062296"},{"key":"2025091715210250715_j_auto-2025-0037_ref_048","unstructured":"S. Kerschbaum, Backstepping Control of Coupled Parabolic Systems with Varying Parameters, Dissertation, Erlangen, FAU University Press, 2021."},{"key":"2025091715210250715_j_auto-2025-0037_ref_049","doi-asserted-by":"crossref","unstructured":"Y. Orlov, \u201cOn general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on \u201cISS with respect to boundary disturbances for 1-D parabolic PDEs\u201d,\u201d IEEE Trans. Autom. Control, vol.\u00a062, pp.\u00a05970\u20135973, 2017. https:\/\/doi.org\/10.1109\/tac.2017.2694425.","DOI":"10.1109\/TAC.2017.2694425"},{"key":"2025091715210250715_j_auto-2025-0037_ref_050","doi-asserted-by":"crossref","unstructured":"M. J. Balas, \u201cTowards a more practical control theory for distributed parameter systems,\u201d in Control and Dynamics Systems: Advances in Theory and Applications, vol.\u00a018, C. T. Leondes, Ed., New York, Academic Press, 1980, pp.\u00a0361\u2013421.","DOI":"10.1016\/B978-0-12-012718-4.50014-1"},{"key":"2025091715210250715_j_auto-2025-0037_ref_051","doi-asserted-by":"crossref","unstructured":"P. Schmid, \u201cDynamic mode decomposition of numerical and experimental data,\u201d J. Fluid Mech., vol.\u00a0656, pp.\u00a05\u201328, 2010. https:\/\/doi.org\/10.1017\/s0022112010001217.","DOI":"10.1017\/S0022112010001217"},{"key":"2025091715210250715_j_auto-2025-0037_ref_052","doi-asserted-by":"crossref","unstructured":"A. Smyshlyaev and M. Krstic, \u201cClosed-form boundary state feedbacks for a class of 1-D partial integro-differential equations,\u201d IEEE Trans. Autom. Control, vol.\u00a049, pp.\u00a02185\u20132202, 2004. https:\/\/doi.org\/10.1109\/tac.2004.838495.","DOI":"10.1109\/TAC.2004.838495"},{"key":"2025091715210250715_j_auto-2025-0037_ref_053","doi-asserted-by":"crossref","unstructured":"D. Bernstein, Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas, Princeton, New Jersey, Princeton University Press, 2018.","DOI":"10.1515\/9781400888252"},{"key":"2025091715210250715_j_auto-2025-0037_ref_054","doi-asserted-by":"crossref","unstructured":"E. Gilbert, \u201cControllability and observability in multivariable control systems,\u201d SIAM J. Control, Ser. A, vol.\u00a01, pp.\u00a0128\u2013151, 1963. https:\/\/doi.org\/10.1137\/0301009.","DOI":"10.1137\/0301009"},{"key":"2025091715210250715_j_auto-2025-0037_ref_055","doi-asserted-by":"crossref","unstructured":"H. Zwart, \u201cTransfer functions for infinite-dimensional systems,\u201d Syst. Control Lett., vol.\u00a052, pp.\u00a0247\u2013255, 2004. https:\/\/doi.org\/10.1016\/j.sysconle.2004.02.002.","DOI":"10.1016\/j.sysconle.2004.02.002"}],"container-title":["at - Automatisierungstechnik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/auto-2025-0037\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/auto-2025-0037\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T15:22:29Z","timestamp":1758122549000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/auto-2025-0037\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,1]]},"references-count":55,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8,6]]},"published-print":{"date-parts":[[2025,8,26]]}},"alternative-id":["10.1515\/auto-2025-0037"],"URL":"https:\/\/doi.org\/10.1515\/auto-2025-0037","relation":{},"ISSN":["0178-2312","2196-677X"],"issn-type":[{"type":"print","value":"0178-2312"},{"type":"electronic","value":"2196-677X"}],"subject":[],"published":{"date-parts":[[2025,8,1]]}}}