Zusammenfassung
In diesem Beitrag wird ein alternatives, datenbasiertes Framework für die Fehlerdiagnose (FDI) basierend auf symmetrischen, positiv-definiten (SPD) Matrizen eingeführt. In der Fehlerdiagnose und Regelungstheorie enthalten SPD-Matrizen abhängig von der Analyse und Interpretation wichtige Informationen über das betrachtete System. Zur Berücksichtigung ihrer besonderen Eigenschaften wird die Riemann’sche Geometrie als mathematische Grundlage genutzt. Das grundlegende SPD-Matrix-basierte FD-Schema erlaubt eine flexible Umsetzung ohne Annahmen der statistischen Verteilung der Daten. Außerdem wird ein Überblick über mögliche Realisierungen des Frameworks für die modell- und datenbasierte FDI sowie im Bereich des Maschinellen Lernens (ML) gegeben. Es wird eine neuartige Modellierung stabiler, linearer zeitinvarianter Systeme vorgestellt und zu einem FD-Schema erweitert.
Abstract
In this work, an alternative, data-driven fault diagnosis (FDI)-framework based on symmetric, positive-definite (SPD) matrices will be introduced. In fault diagnosis and control theory, SPD-matrices serve important information about the system under investigation depending on the analysis and interpretation. Taking into account their special characteristics and for the purpose of an alternative way of investigation, the basic concepts of Riemannian Geometry are introduced as a mathematical tool. The fundamental SPD-matrix-based FD-scheme is characterised by a flexible implementation and independence of statistical distribution of data. The article also delivers an overview about the possible realisations in model-based and data-driven FDI as well as in the area of machine learning (ML). Further, a new modelling of a linear, time-invariant system will be introduced and extended to an FD scheme.
Über die Autoren
Caroline C. Zhu schloss ihr Masterstudium Elektro- und Informationstechnik mit Schwerpunkt Automatisierungstechnik an der Universität Duisburg-Essen im Jahr 2018 ab. Sie ist nun als wissenschaftliche Mitarbeiterin im Fachgebiet Automatisierungstechnik und komplexe Systeme an der Universität Duisburg-Essen tätig. Ihre Forschungsinteressen sind u.a. datenbasierte Fehlerdiagnose und Process Monitoring.
Kristian Kasten schloss sein Maschinenbaustudium an der RWTH Aachen mit dem Schwerpunkt Energietechnik im Jahr 2015 ab. Seitdem entwickelt und implementiert er Überwachungskonzepte zur Steigerung der Verfügbarkeit von verfahrenstechnischen Apparaten und Maschinen innerhalb der BASF SE. Seine Forschungsinteressen sind die datenbasierte Fehlerdiagnose und Restlebenszeitschätzung sowie deren Skalierung im industriellen Umfeld.
Seit seiner Promotion 1992 über nichtlineare Systemdynamik und Regelungstechnik an der Universität Stuttgart ist Joachim Birk in der BASF SE tätig. Er leitet das Fachzentrum Automatisierungstechnik in Ludwigshafen und verantwortet als Vice President und Executive Expert of Automation Technology die Fachgebiete Control Systems, Regulated Automation Solutions, Advanced Process Control, Manufacturing Execution Systems (MES) sowie Automation Security. Darüber hinaus hat er seit 2012 einen Lehrauftrag an der Universität Stuttgart und lehrt als Honorarprofessor verschiedene Themen rund um OT (Operational Technology). 2016 wurde er für sein hohes Engagement und seinen langjährigen Einsatz in der NAMUR mit der Goldenen Ehrennadel der NAMUR ausgezeichnet.
Nach seiner Promotion 1992 an der Universität Duisburg war Steven Ding bei der Firma Rheinmetall GmbH und an der Fachhochschule Lausitz tätig. Seit 2001 ist er Professor der Automatisierungstechnik an der Universität Duisburg-Essen und leitet das Fachgebiet Automatisierungstechnik und komplexe Systeme (AKS).
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Competing interests: The authors state no conflict of interest.
-
Research funding: None declared.
-
Data availability: Not applicable.
Literatur
[1] S. X. Ding, Model-Based Fault Diagnosis Techniques, London, Springer, 2013.10.1007/978-1-4471-4799-2Search in Google Scholar
[2] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems, US, Springer, 2012.Search in Google Scholar
[3] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, Berlin, Heidelberg, Springer, 2016.10.1007/978-3-662-47943-8Search in Google Scholar
[4] E. A. García and P. Frank, “Deterministic nonlinear observer-based approaches to fault diagnosis: a survey,” Control Eng. Pract., vol. 5, no. 5, pp. 663–670, 1997. https://doi.org/10.1016/s0967-0661(97)00048-8.Search in Google Scholar
[5] D. G. Luenberger, “Observing the state of a linear system,” IEEE Trans. Mil. Electron., vol. 8, no. 2, pp. 74–80, 1964. https://doi.org/10.1109/tme.1964.4323124.Search in Google Scholar
[6] P. Zhang and S. X. Ding, “An integrated trade-off design of observer based fault detection systems,” Automatica, vol. 44, no. 7, pp. 1886–1894, 2008. https://doi.org/10.1016/j.automatica.2007.11.021.Search in Google Scholar
[7] Z. Ge, Z. Song, and F. Gao, “Review of recent research on data-based process monitoring,” Ind. Eng. Chem. Res., vol. 52, no. 10, pp. 3543–3562, 2013. https://doi.org/10.1021/ie302069q.Search in Google Scholar
[8] S. Ding, P. Zhang, A. Naik, E. Ding, and B. Huang, “Subspace method aided data-driven design of fault detection and isolation systems,” J. Process Control, vol. 19, no. 9, pp. 1496–1510, 2009. https://doi.org/10.1016/j.jprocont.2009.07.005.Search in Google Scholar
[9] Y. Wang, G. Ma, S. X. Ding, and C. Li, “Subspace aided data-driven design of robust fault detection and isolation systems,” Automatica, vol. 47, no. 11, pp. 2474–2480, 2011. https://doi.org/10.1016/j.automatica.2011.05.028.Search in Google Scholar
[10] B. Huang and R. Kadali, Dynamic Modeling, Predictive Control and Performance Monitoring: A Data-Driven Subspace Approach, London: Springer, 2008.Search in Google Scholar
[11] H. Luo, K. Li, O. Kaynak, S. Yin, M. Huo, and H. Zhao, “A robust data-driven fault detection approach for rolling mills with unknown roll eccentricity,” IEEE Trans. Control Syst. Technol., vol. 28, no. 6, pp. 2641–2648, 2020. https://doi.org/10.1109/tcst.2019.2942799.Search in Google Scholar
[12] S. X. Ding, Advanced Methods for Fault Diagnosis and Fault-Tolerant Control, Berlin, Heidelberg, Springer, 2021.10.1007/978-3-662-62004-5Search in Google Scholar
[13] Z. Chen, S. X. Ding, K. Zhang, Z. Li, and Z. Hu, “Canonical correlation analysis-based fault detection methods with application to alumina evaporation process,” Control Eng. Pract., vol. 46, pp. 51–58, 2016. https://doi.org/10.1016/j.conengprac.2015.10.006.Search in Google Scholar
[14] Z. Chen, et al.., “A just-in-time-learning-aided canonical correlation analysis method for multimode process monitoring and fault detection,” IEEE Trans. Ind. Electron., vol. 68, no. 6, pp. 5259–5270, 2021. https://doi.org/10.1109/tie.2020.2989708.Search in Google Scholar
[15] Y. Tao, H. Shi, B. Song, and S. Tan, “A novel dynamic weight principal component analysis method and hierarchical monitoring strategy for process fault detection and diagnosis,” IEEE Trans. Ind. Electron., vol. 67, no. 9, pp. 7994–8004, 2020. https://doi.org/10.1109/tie.2019.2942560.Search in Google Scholar
[16] L. H. Chiang, E. L. Russell, and R. D. Braatz, “Fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis,” Chemom. Intell. Lab. Syst., vol. 50, no. 2, pp. 243–252, 2000. https://doi.org/10.1016/s0169-7439(99)00061-1.Search in Google Scholar
[17] K. Zhang, K. Peng, and Y. A. W. Shardt, “A comparison of different statistics for detecting multiplicative faults in multivariate statistics-based fault detection approaches,” IEEE Access, vol. 6, pp. 43808–43823, 2018. https://doi.org/10.1109/access.2018.2862940.Search in Google Scholar
[18] M. Kano, S. Tanaka, S. Hasebe, I. Hashimoto, and H. Ohno, “Monitoring independent components for fault detection,” AIChE J., vol. 49, no. 4, pp. 969–976, 2003. https://doi.org/10.1002/aic.690490414.Search in Google Scholar
[19] J. Zeng, U. Kruger, J. Geluk, X. Wang, and L. Xie, “Detecting abnormal situations using the Kullback–Leibler divergence,” Automatica, vol. 50, no. 11, pp. 2777–2786, 2014. https://doi.org/10.1016/j.automatica.2014.09.005.Search in Google Scholar
[20] Z. Ge, Z. Song, S. X. Ding, and B. Huang, “Data mining and analytics in the process industry: the role of machine learning,” IEEE Access, vol. 5, pp. 20590–20616, 2017. https://doi.org/10.1109/access.2017.2756872.Search in Google Scholar
[21] R. Iqbal, T. Maniak, F. Doctor, and C. Karyotis, “Fault detection and isolation in industrial processes using deep learning approaches,” IEEE Trans. Ind. Inform., vol. 15, no. 5, pp. 3077–3084, 2019. https://doi.org/10.1109/tii.2019.2902274.Search in Google Scholar
[22] B. Schölkopf, R. C. Williamson, A. Smola, J. Shawe-Taylor, and J. Platt, “Support vector method for novelty detection,” Adv. Neural Inf. Process. Syst., vol. 12, pp. 582–588, 1999.Search in Google Scholar
[23] S. Mahadevan and S. L. Shah, “Fault detection and diagnosis in process data using one-class support vector machines,” J. Process Control, vol. 19, no. 10, pp. 1627–1639, 2009. https://doi.org/10.1016/j.jprocont.2009.07.011.Search in Google Scholar
[24] C. C. Zhu, L. Li, and S. X. Ding, “Multiplicative fault detection and isolation in dynamic systems using data-driven k-gap metric based kNN algorithm,” IFAC-PapersOnLine, vol. 55, no. 6, pp. 169–174, 2022. https://doi.org/10.1016/j.ifacol.2022.07.124.Search in Google Scholar
[25] L. Li, S. X. Ding, K. Liang, Z. Chen, and T. Xue, “Control theoretically explainable application of autoencoder methods to fault detection in nonlinear dynamic systems,” [Online], 2022. Available at: https://arxiv.org/abs/2208.01291.Search in Google Scholar
[26] H. Yu, S. Yang, S. X. Ding, Z. Dai, and S. Yin, “A data-driven fault detection scheme for complex industrial systems using riemannian metric and randomized algorithms,” in 2020 IEEE 29th International Symposium on Industrial Electronics (ISIE), IEEE, 2020.10.1109/ISIE45063.2020.9152552Search in Google Scholar
[27] J. Lunze, Regelungstechnik 2, Berlin, Heidelberg, Springer, 2016.10.1007/978-3-662-52676-7Search in Google Scholar
[28] L. H. Chiang, E. L. Russell, and R. D. Braatz, Fault Detection and Diagnosis in Industrial Systems, London, Springer, 2001.10.1007/978-1-4471-0347-9Search in Google Scholar
[29] S. X. Ding, Data-driven Design of Fault Diagnosis and Fault-Tolerant Control Systems, London, Springer, 2014.10.1007/978-1-4471-6410-4Search in Google Scholar
[30] S. J. Qin, “Statistical process monitoring: basics and beyond,” J. Chemom., vol. 17, nos. 8–9, pp. 480–502, 2003. https://doi.org/10.1002/cem.800.Search in Google Scholar
[31] R. De Maesschalck, D. Jouan-Rimbaud, and D. Massart, “The Mahalanobis distance,” Chemom. Intell. Lab. Syst., vol. 50, no. 1, pp. 1–18, 2000. https://doi.org/10.1016/s0169-7439(99)00047-7.Search in Google Scholar
[32] J. MacGregor and T. Kourti, “Statistical process control of multivariate processes,” Control Eng. Pract., vol. 3, no. 3, pp. 403–414, 1995. https://doi.org/10.1016/0967-0661(95)00014-l.Search in Google Scholar
[33] E. L. Russell, L. H. Chiang, and R. D. Braatz, “Fault detection in industrial processes using canonical variate analysis and dynamic principal component analysis,” Chemom. Intell. Lab. Syst., vol. 51, no. 1, pp. 81–93, 2000. https://doi.org/10.1016/s0169-7439(00)00058-7.Search in Google Scholar
[34] S. J. Qin, “Recursive PLS algorithms for adaptive data modeling,” Comput. Chem. Eng., vol. 22, nos. 4–5, pp. 503–514, 1998. https://doi.org/10.1016/s0098-1354(97)00262-7.Search in Google Scholar
[35] K. Zhang, S. X. Ding, Y. A. W. Shardt, Z. Chen, and K. Peng, “Assessment of T2- and Q-statistics for detecting additive and multiplicative faults in multivariate statistical process monitoring,” J. Frank. Inst., vol. 354, no. 2, pp. 668–688, 2017. https://doi.org/10.1016/j.jfranklin.2016.10.033.Search in Google Scholar
[36] J.-J. E. Slotine and W. Li, Applied Nonlinear Control, vol. 199, Englewood Cliffs, NJ, Prentice Hall, 1991.Search in Google Scholar
[37] K. Zhou and J. C. Doyle, Essentials of Robust Control, Upper Saddle River, NJ, Prentice Hall, 1998.Search in Google Scholar
[38] R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng., vol. 82, no. 1, pp. 35–45, 1960. https://doi.org/10.1115/1.3662552.Search in Google Scholar
[39] S. X. Ding, “A note on diagnosis and performance degradation detection in automatic control systems towards functional safety and cyber security,” Security and Safety, vol. 1, no. 2022004, p. 2022004, 2022. https://doi.org/10.1051/sands/2022004.Search in Google Scholar
[40] M. P. do Carmo, Riemannian Geometry: Theory & Applications, Boston, Birkhäuser, 1992.10.1007/978-1-4757-2201-7Search in Google Scholar
[41] S. Amari, Information Geometry and its Applications, Japan, Springer, 2016.10.1007/978-4-431-55978-8Search in Google Scholar
[42] R. A. Horn and C. R. Johnson, Matrix analysis, 2nd ed. Cambridge, UK, Cambridge Univ. Press, 2013.Search in Google Scholar
[43] X. Pennec, P. Fillard, and N. Ayache, “A riemannian framework for tensor computing,” Int. J. Comput. Vis., vol. 66, no. 1, pp. 41–66, 2006. https://doi.org/10.1007/s11263-005-3222-z.Search in Google Scholar
[44] P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton, NJ, USA, Princeton University Press, 2008.10.1515/9781400830244Search in Google Scholar
[45] R. Bhatia, Positive Definite Matrices, Princeton, NJ, USA, Princeton University Press, 2009.10.1515/9781400827787Search in Google Scholar
[46] P. T. Fletcher, “Geodesic regression and the theory of least squares on riemannian manifolds,” Int. J. Comput. Vis., vol. 105, no. 2, pp. 171–185, 2012. https://doi.org/10.1007/s11263-012-0591-y.Search in Google Scholar
[47] O. Yair, M. Ben-Chen, and R. Talmon, “Parallel transport on the cone manifold of SPD matrices for domain adaptation,” IEEE Trans. Signal Process., vol. 67, no. 7, pp. 1797–1811, 2019. https://doi.org/10.1109/tsp.2019.2894801.Search in Google Scholar
[48] S. Yin, S. X. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,” J. Process Control, vol. 22, no. 9, pp. 1567–1581, 2012. https://doi.org/10.1016/j.jprocont.2012.06.009.Search in Google Scholar
[49] M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005. https://doi.org/10.1137/s0895479803436937.Search in Google Scholar
[50] R. Bhatia and J. Holbrook, “Riemannian geometry and matrix geometric means,” Linear Algebra Appl., vol. 413, nos. 2–3, pp. 594–618, 2006. https://doi.org/10.1016/j.laa.2005.08.025.Search in Google Scholar
[51] D. A. Bini and B. Iannazzo, “Computing the Karcher mean of symmetric positive definite matrices,” Linear Algebra Appl., vol. 438, no. 4, pp. 1700–1710, 2013. https://doi.org/10.1016/j.laa.2011.08.052.Search in Google Scholar
[52] B. Ng, G. Varoquaux, J. B. Poline, M. Greicius, and B. Thirion, “Transport on riemannian manifold for connectivity-based brain decoding,” IEEE Trans. Med. Imaging, vol. 35, no. 1, pp. 208–216, 2016. https://doi.org/10.1109/tmi.2015.2463723.Search in Google Scholar
[53] P. D. J. Lunze, Regelungstechnik 1: Systemtheoretische Grundlagen, Analyse und Entwurf einschleifiger Regelungen, Berlin, Heidelberg, Springer, 2020.10.1007/978-3-662-60746-6Search in Google Scholar
[54] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, New York, NY, USA, Wiley John + Sons, 2000.Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
