Numerical Methods for Monitoring Rare Events in Nonlinear Stochastic Systems
https://doi.org/10.17587/mau.22.291-297
Abstract
In this article, we consider the development of numerical methods of large deviations analysis for rare events in nonlinear stochastic systems. The large deviations of the controlled process from a certain stable state are the basis for predicting the occurrenceof a critical situation (a rare event). The rare event forecasting problem is reduced to the Lagrange-Pontryagin optimal control problem.The presented approach for solving the Lagrange-Pontryagin problem differs from the approach used earlier for linear systems in that it uses feedback control. In the nonlinear case, approximate methods based on the representation of the system model in the state-space form with state-dependent coefficients (SDC) matrixes are used: the state-dependent Riccati equation (SDRE) and the asymptotic sequence of Riccati equations (ASRE). The considered optimal control problem allow us to obtain a numerical-analytical solutionthat is convenient for real-time implementation. Based on the developed methods of large deviations analysis, algorithms for estimating the probability of occurrence of a rare event in a dynamical systemare presented. The numerical applicability of the developed methods is shown by the example of the FitzHugh-Nagumo model for the analysis of switching between excitable modes. The simulation results revealed an additional problem related to the so-called parameterization problem of the SDC matrices. Since the use of different representations for SDC matrices gives different results in terms of the system trajectory, the choice of matrices is proposed to be carried out at each algorithm iteration so as to provide conditions for the solvability of the Lagrange-Pontryagin problem.
Keywords
About the Authors
A. A. KabanovRussian Federation
Ph.D., Associate Professor
Sevastopol, 299053
S. A. Dubovik
Russian Federation
Sevastopol, 299053
References
1. Grafke T., Vanden-Eijnden E. Numerical computation of rare events via large deviation theory, Chaos,2019, vol. 29, paper no. 063118.
2. Sapsis T. P. New perspectives for the prediction and statistical quantification of extreme events in high-dimensional dynamical systems, Philosophical Transactions of the Royal Society A,2018, vol. 376, paper no. 20170133.
3. Dubovik S. A., Kabanov A. A. Profiles of critical states in diagnostics of controlled processes, MATEC Web of Conferences,2018, vol. 224, paper no. 04024.
4. Dubovik S. A. Asymptotic semantization of data in control systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2019, vol. 20, no. 8, pp. 461—471 (In Russian).
5. Kabanov A. A., Dubovik S. A. Methods of modeling and probabilistic analysis of large deviations of dynamic systems, Journal of Physics: Conference Series, 2020, vol. 1661, paper no. 012044.
6. Çimen T. State-dependent Riccati equation (SDRE) control: a survey, IFAC Proceedings Volumes, 2008, vol. 41, pp. 3761—3775.
7. Çimen T. Survey of state-dependent Riccati equation in nonlinear optimal feedback control synthesis, Journal of Guidance, Control, and Dynamics, 2012, vol. 35, pp. 1025—1047.
8. Çimen T., Banks S. P. Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria, Systems & Control Letters, 2004, vol. 53, pp. 327—346.
9. Topputo F., Miani M., Bernelli-Zazzera F. Optimal selection of the coefficient matrix in state-dependent control methods, Journal of Guidance, Control, and Dynamics, 2015, vol. 38, pp. 861—873.
10. Heydari A., Balakrishnan S. N. Closed-form solution to finite-horizon suboptimal control of nonlinear systems, International Journal of Robust and Nonlinear Control, 2014, vol. 25, pp. 2687—2704.
11. Izhikevich E. M. Dynamical systems in neuroscience: the geometry of excitability and bursting, Cambridge, Computational neuroscience, MIT Press, 2007, 464 p.
12. Bashkirtseva I., Ryashko L. Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique, Physical Review E, 2011, vol. 83, paper no. 061109.
13. Doss C., Thieullen M. Oscillations and random perturbations of a FitzHugh-Nagumo system, Preprint, arXiv:0906.2671v1, 2009.
14. Freidlin M. I., Wentzell A. D. Random perturbations of dynamical systems, 3. ed., Heidelberg, Grundlehren der mathematischen Wissenschaften, Springer, 2012, 460 p.
15. Bryson A. E., Ho Y.-C. Applied optimal control: optimization, estimation, and control, New York, Taylor & Francis, 1975, 482 p.
16. Bachar M., Batzel J., Ditlevsen S. Stochastic biomathematical models, Lecture Notes in Mathematics, Berlin, Springer Berlin Heidelberg, 2013, 206 p.
17. Heinrich M., Dahms T., Flunkert V., Teitsworth S. W., Schöll E.Symmetry-breaking transitions in networks of nonlinear circuit elements, New Journal of Physics, 2010, vol. 12, paper no. 113030.
18. Nash M. P., Panfilov A. V. Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progress in Biophysics and Molecular Biology, 2004, vol. 85, pp. 501—522.
19. Ganopolski A., Rahmstorf S. Abrupt glacial climate changes due to stochastic resonance, Physical Review Letters, 2002, vol.88, paper no. 038501.
20. Plotnikov S. A., Fradkov A. L. Controlled synchronization in two hybrid FitzHugh-Nagumo systems, IFAC-PapersOnLine, 2016, vol. 49, iss. 14, pp. 137—141.
Review
For citations:
Kabanov A.A., Dubovik S.A. Numerical Methods for Monitoring Rare Events in Nonlinear Stochastic Systems. Mekhatronika, Avtomatizatsiya, Upravlenie. 2021;22(6):291-297. (In Russ.) https://doi.org/10.17587/mau.22.291-297