Control PID estadístico para la regulación de variables en un modelo de Lorenz

Jesús Eduardo Rodríguez Bravo; José Manuel Cruz Olguín; Anilu Franco Arcega; Manuel Alejandro Ojeda-Misses; Fernando Hernández Morales

doi:10.29057/icbi.v14iEspecial2.16653

Autores/as

Jesús Eduardo Rodríguez Bravo Universidad Autónoma del Estado de Hidalgo https://orcid.org/0009-0000-3309-3344
José Manuel Cruz Olguín Universidad Autónoma del Estado de Hidalgo https://orcid.org/0009-0008-1328-7780
Anilu Franco Arcega Universidad Autónoma del Estado de Hidalgo https://orcid.org/0000-0002-9415-8313
Manuel Alejandro Ojeda-Misses Universidad Autónoma del Estado de Hidalgo https://orcid.org/0000-0003-3963-5399
Fernando Hernández Morales Universidad Autónoma del Estado de Hidalgo https://orcid.org/0009-0007-6276-7548

DOI:

https://doi.org/10.29057/icbi.v14iEspecial2.16653

Palabras clave:

caos, controlador PID, control estadístico, estabilización, Modelo de Lorenz

Resumen

Este artículo presenta la regulación de un sistema caótico de Lorenz mediante dos estrategias de control, un controlador proporcional (P) y un controlador Proporcional-Integral-Derivativo (PID) con un algoritmo de sintonización basado en adaptación estadística. A diferencia de los métodos convencionales que requieren un modelo matemático preciso, la propuesta ajusta las ganancias mediante la normalización del error, usando la media y la desviación estándar para estimar la probabilidad del error y una función de mapeo, que permite ajustar las ganancias en tiempo real. Este enfoque ayuda en la robustez del controlador reduciendo las oscilaciones. Los resultados de simulación muestran que el control proporcional sólo atenúa parcialmente las oscilaciones del sistema de Lorenz, mientras que el PID estadístico logra estabilizar los estados x,y,z hacia la referencia deseada, controlando el atractor caótico con menores oscilaciones. Finalmente, se presentan las señales de error, los errores cuadráticos medios y las señales de control, confirmando el desempeño del controlador estadístico en sistemas complejos.

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Citas

Bitseki Penda, S. V., & Delmas, J.-F. (2024). Central limit theorem for bifurcating Markov chains. Statistics & Probability Letters. https://doi.org/10.1080/17442508.2023.2295847

Buizza, R. (2018). Introduction to the special issue on “25 years of ensemble forecasting”. Quarterly Journal of the Royal Meteorological Society, 144(S1), 1–2. https://doi.org/10.1002/qj.3170

Chakraborty, A., & Veeresha, P. (2024). Effects of global warming, time delay and chaos control on the dynamics of a chaotic atmospheric propagation model within the frame of Caputo fractional operator. Communications in Nonlinear Science and Numerical Simulation, 128, 107657. https://doi.org/10.1016/j.cnsns.2023.107657

Oestreicher, C. (2007). A history of chaos theory. Dialogues in Clinical Neuroscience, 9(3), 279–289.

Ojeda-Misses, M. A., Martines-Arano, H., López-Morales, V., Franco-Árcega, A., & Márquez-Grajales, A. (2024). Self-tuned closed-loop controller based on statistical data using a servomechanism. In 2024 XXVI Robotics Mexican Congress (COMRob), Torreón, Coahuila, Mexico, pp. 27–32. https://doi.org/10.1109/COMRob64055.2024.10777440

Ojeda-Misses, M., Martines-Arano, H., Sampedro-Mendoza, A., Franco-Árcega, A., & López-Morales, V. (2025). Diseño de un controlador mediante datos estadísticos en lazo cerrado para un servomecanismo mediante una técnica de autosintonización. RIIIT Revista Internacional de Investigación e Innovación Tecnológica, 12(72), 24–43. https://revistas.uadec.mx/index.php/RIIIT/article/view/104

Ott, E., Grebogi, C., & Yorke, J. A. (1990). Controlling chaos. Physical Review Letters, 64(11), 1196–1199. https://doi.org/10.1103/PhysRevLett.64.1196

Palmer, T. N., Doblas-Reyes, F. J., Hagedorn, R., & Weisheimer, A. (2005). Probabilistic prediction of climate using multi-model ensembles: From basics to applications. Philosophical Transactions of the Royal Society B, 360(1463), 1991–1998. https://doi.org/10.1098/rstb.2005.1750

Pandey, S., & Schumacher, J. (2021). Short- and long-term predictions of chaotic flows and extreme events: A physics-constrained reservoir computing approach. Proceedings of the Royal Society A, 477(2253). https://doi.org/10.1098/rspa.2021.0135

Pasini, A., & Pelino, V. (2005). Can we estimate atmospheric predictability by performance of neural network forecasting? The toy case studies of unforced and forced Lorenz models. In CIMSA 2005 IEEE International Conference on Computational Intelligence for Measurement Systems and Applications, 69–74. https://doi.org/10.1109/CIMSA.2005.1522829

Pathak, J., Lu, Z., Hunt, B., Girvan, M., & Ott, E. (2020). Using reservoir computing to predict and prevent extreme events. Physical Review E, 101(2), 022209. https://doi.org/10.1103/PhysRevE.101.022209

Penny, S., Smith, T., Chen, T. C., Platt, J., Lin, H. Y., Goodliff, M., & Abarbanel, H. D. (2023). Application of recurrent neural networks to model bias correction: Idealized experiments with the Lorenz-96 model. Journal of Advances in Modeling Earth Systems, 15(1), e2022MS003164. https://doi.org/10.1029/2022MS003164

Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Physics Letters A, 170(6), 421–428. https://doi.org/10.1016/0375-9601(92)90745-8

Shen, B. L., Wang, M., Yan, P., Yu, H., Song, J., & Da, C. J. (2018). Stable and unstable regions of the Lorenz system. Scientific Reports, 8(1). https://doi.org/10.1038/s41598-018-33010

Sparrow, C. (1982). The Lorenz equations: Bifurcations, chaos, and strange attractors. Springer-Verlag.

Sun, Q., Miyoshi, T., & Richard, S. (2023). Control simulation experiments of extreme events with the Lorenz-96 model. Nonlinear Processes in Geophysics, 30(2), 117–128. https://doi.org/10.5194/npg-30-117-2023

Uy, W. I. T., Grigoriu, M. D., & Juniper, M. P. (2024). Uncertainty quantification of time-average quantities of chaotic systems using sensitivity-enhanced polynomial chaos expansion. Physical Review E, 109(4), 044208. https://doi.org/10.1103/PhysRevE.109.044208

Wang, B., Ding, H., Zhang, S., & Chen, Y. (2021). Robust synchronization of class chaotic systems using novel time-varying gain disturbance observer-based sliding mode control. Complexity, 2021, 8845553. https://doi.org/10.1155/2021/8845553

Wang, H., Dong, G., & Chen, C. (2018). Hopf bifurcation control in a Lorenz type system. International Journal of Bifurcation and Chaos, 28(10), 1850127. https://doi.org/10.1142/S0218127418501274

Chattopadhyay, A., Hassanzadeh, P., & Subramanian, D. (2020). Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods. Nonlinear Processes in Geophysics, 27(3), 373–389. https://doi.org/10.5194/npg-27-373-2020

Chen, W. H., Wei, D., & Lu, X. (2014). Lyapunov-based controller for a class of stochastic chaotic systems. Advances in Computer Science and its Applications, 2014, 613463. https://doi.org/10.1155/2014/613463

Chou, J. F., Zheng, Z. H., & Sun, S. P. (2010). The think about 10–30 d extended-range numerical weather prediction strategy—facing the atmosphere chaos. Scientia Meteorologica Sinica, 30(5), 569–573.

Curry, J. H. (1978). A generalized Lorenz system. Communications in Mathematical Physics, 60(3), 193–204.

Dhooge, A., Govaerts, W., Kuznetsov, Y. A., Meijer, H. G. E., & Sautois, B. (2024). Computational bifurcation analysis. arXiv preprint arXiv:2411.00735. https://doi.org/10.48550/arXiv.2411.00735

Díaz Rojas, C. A., & Pino, R. (2023). Principios de formación de estrategias complejas: un enfoque basado en la teoría de la complejidad. 360: Revista de Ciencias de la Gestión, 8(8).

Ding, R., Liu, B., Gu, B., Li, J., & Li, X. (2019). Predictability of ensemble forecasting estimated using the Kullback–Leibler divergence in the Lorenz model. Advances in Atmospheric Sciences, 36(8), 837–846. https://doi.org/10.1007/s00376-019-9034-9

Harter, F. P., Yamasaki, Y., & Beck, V. C. (2015). Assimilação de dados via método 3D-Var em dinâmica caótica do modelo de Lorenz. Anuário do Instituto de Geociências - UFRJ, 38(1), 73–80. https://doi.org/10.11137/2015_1_73_80

Krishnamurti, T. N., & Ramanathan, Y. (1982). Sensitivity of the monsoon onset to differential heating. Journal of the Atmospheric Sciences, 39(6), 1290–1306.

Liao, S., & Wang, P. (2014). On the mathematically reliable long-term simulation of chaotic solutions of Lorenz equation in the interval [0,10000]. Science China Physics, Mechanics & Astronomy, 57(2), 330–335. https://doi.org/10.1007/s11433-013-5375-z

Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.

Milinski, S., Mahlstein, I., & Gromer, D. (2021). An ensemble-based statistical methodology to detect differences in weather and climate model.