Seyrek Tanılama Yöntemi ile Doğrusal Olmayan Dinamik Sistemlerin Model İncelenmesi

Yıl 2020, Ejosat Özel Sayı 2020 (ISMSIT), 254 - 263, 30.11.2020

Nezir Kadah , Necdet Sinan Özbek

https://doi.org/10.31590/ejosat.822361

Öz

Doğrusal olmayan sistemleri tanımlamak için seyrek regresyon tekniklerine dayanan doğrusal olmayan dinamiklerin seyrek tanımlanması (SINDy) son yıllarda ortaya konan veriye dayalı model tanımlama yöntemlerinden biridir. Sistem tanılamada sistemin model denklemleri verilerden çıkarılır. Mühendislik, sağlık hizmetleri ve ekonomi bilimlerinin çoğundan yeterli veri mevcut olmasına rağmen, sistem davranışını temsil eden çok az sayıda iyi tanımlanmış model vardır. Sistemin davranışı, veriye dayalı yöntemlerden de tahmin edilebilir. Bu motivasyon göz önünde bulundurularak, bu çalışma doğrusal olmayan sistemlerin matematiksel modelini oluşturmak için çevrimdışı veri odaklı tanımlama tekniklerini ele alır. Doğrusal olmayan sistemlerin veriye dayalı seyrek tanımlanması bir dizi örnekle detaylandırılır. Tanımlama işleminin performansı, gürültülü ölçümlerin varlığında bir takım nicel ölçümler üzerinden tartışılır.

Anahtar Kelimeler

Doğrusal Olmayan Sistemler, Seyrek Tanılama, Seyrek Regresyon, Model Keşfi, Sistem Tanılama

Model Investigation of Nonlinear Dynamical Systems by Sparse Identification

Öz

The sparse identification of nonlinear dynamics (SINDy), which is based on the sparse regression techniques to identify the nonlinear systems, is one of the recent data-driven model identification methods. The model equations of the system are extracted from the data. Although sufficient data is available from most of the engineering, healthcare, and economic sciences, there are few well-defined models to represent the system behaviour that can also be estimated from data-driven methods. With this motivation in mind, this study presents offline data-driven identification techniques to build the mathematical model of nonlinear systems. The data-based sparse identification of nonlinear systems is elaborated with a number of examples. The performance of the identification procedure is discussed in terms of quantitative metrics in the presence of noisy measurements.

Anahtar Kelimeler

Nonlinear System, Sparse Identification, Sparse Regression, Model Discovery, System Identification

Kaynakça

• Ayyad, A., Chehadeh, M., Awad, M., & Zweiri, Y. (2020). Real-Time System Identification Using Deep Learning for Linear Processes With Application to Unmanned Aerial Vehicles. IEEE Access, 8, 122539–122553. https://doi.org/10.1109/ACCESS.2020.3006277
• Bhadriraju, B., Narasingam, A., & Kwon, J. S. Il. (2019). Machine learning-based adaptive model identification of systems: Application to a chemical process. Chemical Engineering Research and Design, 152, 372–383. https://doi.org/10.1016/j.cherd.2019.09.009
• Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Kutz, J. N. (2017). Chaos as an Intermittently Forced Linear System. Nature Communications, 8(19), 34. http://faculty.washington.edu/sbrunton/HAVOK.zip
• Brunton, S. L., & Kutz, J. N. (2019). Data-Driven Science and Engineering: Machine Learning, Dynamical Systems and Control. In Cambridge University Press. Cambridge University Press. https://doi.org/10.1017/9781108380690
• Brunton, S. L., & Nathan Kutz, J. (2019). Methods for data-driven multiscale model discovery for materials. J. Phys.: Mater, 2, 44002. https://doi.org/10.1088/2515-7639/ab291e
• Brunton, S. L., Proctor, J. L., & Nathan Kutz, J. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. PNAS, 113(15). https://doi.org/10.1073/pnas.1517384113
• Calafiore, G. C., El Ghaoui, L. M., & Novara, C. (2015). Sparse identification of posynomial models. Automatica, 59, 27–34. https://doi.org/10.1016/j.automatica.2015.06.003
• Callaham, J. L., Maeda, K., & Brunton, S. L. (2019). Robust flow reconstruction from limited measurements via sparse representation. PHYSICAL REVIEW FLUIDS, 4, 103907. https://doi.org/10.1103/PhysRevFluids.4.103907
• Champion, K. P., Brunton \ddagger, S. L., & Nathan Kutz, J. (2019). Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings. SIAM J. APPLIED DYNAMICAL SYSTEMS, 18(1), 312–333. https://doi.org/10.1137/18M1188227
• Chartrand, R. (2011). Numerical Differentiation of Noisy, Nonsmooth Data. ISRN Applied Mathematics, 2011, 1–11. https://doi.org/10.5402/2011/164564
• Chu, H. K., & Hayashibe, M. (2020). Discovering Interpretable Dynamics by Sparsity Promotion on Energy and the Lagrangian. IEEE Robotics and Automation Letters, 5(2), 2154–2160. https://doi.org/10.1109/LRA.2020.2970626
• Corbetta, M. (2020). Application of sparse identification of nonlinear dynamics for physics-informed learning. 2020 IEEE Aerospace Conference, 1–8. https://doi.org/10.1109/aero47225.2020.9172386
• Cortiella, A., Park, K.-C., & Doostan, A. (2020). Sparse Identification of Nonlinear Dynamical Systems via Reweighted $\ell_1$-regularized Least Squares. http://arxiv.org/abs/2005.13232
• De Silva, B. M., Callaham, J., Jonker, J., Goebel, N., Klemisch, J., Mcdonald, D., Hicks, N., Nathan Kutz, J., Brunton, S. L., & Aravkin, A. Y. (2020). Physics-informed machine learning for sensor fault detection with flight test data. 21.
• De Silva, B. M., Higdon, D. M., Brunton, S. L., & Kutz, J. N. (2020). Discovery of Physics From Data: Universal Laws and Discrepancies. Frontiers in Artificial Intelligence, 3(25), 17. https://doi.org/10.3389/frai.2020.00025
• Fey, A., Thiele, G., & Krüger, J. (2020). System identification of a hysteresis-controlled chiller plant using SINDy. 8. http://arxiv.org/abs/2003.07465
• Ford, W. (2014). Numerical Linear Algebra with Applications: Using MATLAB. In Academic Press. Elsevier Inc. https://doi.org/10.1016/C2011-0-07533-6
• Goharoodi, S. K., Dekemele, K., Dupre, L., Loccufier, M., & Crevecoeur, G. (2018). Sparse Identification of Nonlinear Duffing Oscillator From Measurement Data. IFAC-PapersOnLine, 51(33), 162–167. https://doi.org/10.1016/j.ifacol.2018.12.111
• Horrocks, J., & Bauch, C. T. (2020). Algorithmic discovery of dynamic models from infectious disease data. Scientific Reports, 10(1), 1–18. https://doi.org/10.1038/s41598-020-63877-w
• Jain, P., & Pachori, R. B. (2014). Event-Based Method for Instantaneous Fundamental Frequency Estimation from Voiced Speech Based on Eigenvalue Decomposition of the Hankel matrix. IEEE Transactions on Audio, Speech and Language Processing, 22(10), 1467–1482. https://doi.org/10.1109/TASLP.2014.2335056
• Jain, P., & Pachori, R. B. (2015). An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix. Journal of the Franklin Institute, 352(10), 4017–4044. https://doi.org/10.1016/j.jfranklin.2015.05.038
• Kadah, N. (2019). Doğrusal Olmayan RLC Devrelerinin Kararlılık ve Pasiflik Analizi. Van Yuzuncu Yil University. Kaheman, K., Kutz, J. N., & Brunton, S. L. (2020). SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics. http://arxiv.org/abs/2004.02322
• Kaiser, E., Nathan Kutz, J., & Brunton, S. L. (2018). Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences, 474(2219), 14. https://doi.org/https://doi.org/10.1098/rspa.2018.0335
• Kukreja, S. L., Löfberg, J., & Brenner, M. J. (2006). a Least Absolute Shrinkage and Selection Operator (Lasso) for Nonlinear System Identification. IFAC Proceedings Volumes, 39(1), 814–819. https://doi.org/10.3182/20060329-3-au-2901.00128
• Li, H., Wang, Z., & Wang, W. (2020). A Local Sparse Screening Identification Algorithm with Applications. Computer Modeling in Engineering & Sciences, 124(2), 765–782. https://doi.org/10.32604/cmes.2020.010061
• Li, J., & Li, X. (2020). Online sparse identification for regression models. Systems and Control Letters, 141, 104710. https://doi.org/10.1016/j.sysconle.2020.104710
• Lim, R. K., Phan, M. Q., & Longman, R. W. (1998). State-Space System Identification with Identified Hankel Matrix. Department of Mechanical and Aerospace Engineering Technical Report, 3045, 1–36.
• Ljung, L. (2010). Perspectives on system identification. Annual Reviews in Control, 34(1), 1–12. https://doi.org/10.1016/j.arcontrol.2009.12.001
• Lusch, B., Kutz, J. N., & Brunton, S. L. (2018). Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications, 9(1), 4950. https://doi.org/10.1038/s41467-018-07210-0
• Maheshwari, J., Jariwala, R., Pradhan, S., & George, N. V. (2018). Online Least Angle Regression Algorithm for Sparse System Identification. 2017 IEEE International Symposium on Signal Processing and Information Technology, ISSPIT 2017, 191–195. https://doi.org/10.1109/ISSPIT.2017.8388640
• Misra, S., Li, H., & He, J. (2020). Robust geomechanical characterization by analyzing the performance of shallow-learning regression methods using unsupervised clustering methods. In Machine Learning for Subsurface Characterization (pp. 129–155). Elsevier Inc. https://doi.org/10.1016/b978-0-12-817736-5.00005-3
• Niall M. Mangan, Steven L. Brunton, Member, Joshua L. Proctor, and J. N. K. (2016). Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics. IEEE Transactions on Molecular, Biological and Multi-Scale Communications, 22(1), 12. https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=7809160
• Quade, M., Abel, M., Nathan Kutz, J., & Brunton, S. L. (2018). Sparse Identification of Nonlinear Dynamics for Rapid Model Recovery. Chaos: An Interdisciplinary Journal of Nonlinear Science, 10. https://github.com/Ohjeah/sparsereg
• Ranković, V., Radulović, J., Grujović, N., & Divac, D. (2012). Neural Network Model Predictive Control of Nonlinear Systems Using Genetic Algorithms. International Journal of Computers, Communications and Control, 7(3), 540–549. https://doi.org/10.15837/ijccc.2012.3.1394
• Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4). http://advances.sciencemag.org/
• Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society, 58(1), 267–288.
• Wen, H. X., Yang, S. Q., Hong, Y. Q., & Luo, H. (2020). A Partial Update Adaptive Algorithm for Sparse System Identification. IEEE/ACM Transactions on Audio Speech and Language Processing, 28, 240–255. https://doi.org/10.1109/TASLP.2019.2949928
• Zucatti, V., Lui, H. F. S., Pitz, D. B., & Wolf, W. R. (2020). Assessment of reduced-order modeling strategies for convective heat transfer. Numerical Heat Transfer; Part A: Applications, 77(7), 702–729. https://doi.org/10.1080/10407782.2020.1714330