doi: 10.17586/2226-1494-2023-23-5-886-893

Lyapunov function search method for analysis of nonlinear systems stability using genetic algorithm 

A. M. Zenkin, A. A. Peregudin, A. A. Bobtsov

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Zenkin A.M., Peregudin A.A., Bobtsov A.A. Lyapunov function search method for analysis of nonlinear systems stability using genetic algorithm. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 5, pp. 886–893 (in Russian). doi: 10.17586/2226-1494-2023-23-5-886-893

A wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable state vector is considered. The problem of finding a special function (Lyapunov function), which guarantees asymptotic stability for the presented class of nonlinear systems in the framework of the second Lyapunov method, is posed. It is known that the search for the Lyapunov function is an extremely difficult problem that has no universal solution in stability theory. The methods of selection or search of the Lyapunov function for stability analysis of closed linear stationary systems and for nonlinear objects with explicitly expressed linear dynamical and nonlinear static parts are well studied. At the same time, no universal approaches to finding the Lyapunov function for a more general class of nonlinear systems have been identified. In this paper, we propose a new approach to the search of the Lyapunov function for analyzing the stability of smooth continuous dynamic nonlinear systems with a measurable state vector. The essence of the proposed approach consists in the representation of some function through the sum of nonlinear summands representing the elements of the object state vector multiplied by unknown coefficients. The search for these coefficients is performed using a classical genetic algorithm including mutation, selection, and crossover operations. The found coefficients provide all the necessary conditions for the Lyapunov function (within the framework of the second Lyapunov method). The genetic algorithm approach does not require a training sample which imposes restrictions in the form of the structure of control objects included in it. A new method for finding the Lyapunov function represented as a nonlinear series with known functions multiplied by unknown coefficients is proposed. The effectiveness of the proposed method is demonstrated using computer simulations with a fixed number of iterations and varying population size. The dependence of the number of successfully found Lyapunov functions on the number of iterations of the genetic algorithm has been established. The convergence of the genetic algorithm using Holland’s schemes is analyzed. It is shown that the values of the sought coefficients of the potential Lyapunov function, at each algorithm iteration, approach the coefficients of the Lyapunov function which was also represented as a Taylor series. The method proposed in this paper outperforms known analogs in terms of speed, considers the decomposition of the potential Lyapunov function into a Taylor series with unknown coefficients, instead of using counterexamples or template functions.

Keywords: Lyapunov function, mathematical stability, machine learning, genetic algorithm, value function, mathematical pendulum

Acknowledgements. The study was supported by the Russian Science Foundation grant (Project 23-16-00224).

  1. Li X., Yang X. Lyapunov stability analysis for nonlinear systems with state-dependent state delay. Automatica, 2020, vol. 112, pp. 108674–108680.
  2. Lyapunov A.M. The General Problem of the Stability of Motion. Taylor & Francis, 1992, 270 p.
  3. Miroshnik I.V., Nikiforov V.O., Fradkov A.L. Nonlinear Adaptive Control of Complex Dynamic Systems. St. Petersburg, Nauka Publ., 2000, 549 p. (in Russian)
  4. Wu Y., Xie X. A new analysis approach to the output constraint and its application in high-order nonlinear systems. Science China Information Sciences, 2023, vol. 66, no. 5, pp. 159206.
  5. Ivanov S.E., Televnoy A.D. Numerical-analytical transformation method of analyzing nonlinear mathematical models with polynomial structure. Vestnik of Astrakhan State Technical University. Series: Management, Computer Science and Informatics, 2022, no. 2, pp. 97–109. (in Russian).
  6. Srivastava Y., Srivastava S., Chaudhary D., Blanco Valencia X.P. Performance improvement and Lyapunov stability analysis of nonlinear systems using hybrid optimization techniques. Expert Systems, 2022, pp. e13140. in press.
  7. Giesl P., Hafstein S. Computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - Series B, 2015, vol. 20, no. 8, pp. i-ii.
  8. Papachristodoulou A., Anderson J., Valmorbida G., Prajna S., Seiler P., Parrilo P., Peet M.M., Jagt D. SOSTOOLS Version 4.00 Sum of Squares Optimization Toolbox for MATLAB. arXiv, 2013, arXiv:1310.4716.
  9. Hernández-Solano Y., Atencia M. Numerical methods that preserve a Lyapunov function for ordinary differential equations. Mathematics, 2022, vol. 11, no. 1, pp. 71.
  10. Hafstein S., Giesl P. Review on computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - Series B, 2015, vol. 20, no. 8, pp. 2291–2331.
  11. Verdier C.F., Mazo M. Formal synthesis of analytic controllers for sampled-data systems via genetic programming. Proc. of the 2018 IEEE Conference on Decision and Control (CDC), 2018, pp. 4896–4901.
  12. Ben Sassi M.A., Sankaranarayanan S., Chen X., Ábrahám E. Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis. IMA Journal of Mathematical Control and Information, 2016, vol. 33, no. 3, pp. 723–756.
  13. Borodin I.D. The method of statistical synthesis of the Lyapunov function for the study of the stability of an artificial Earth satellite. Vestnik UGATU, 2022, vol. 26, no. 3(97), pp. 14–23. (in Russian).
  14. Feofilov S.V., Kozyr A.V., Khapkin D.L. Synthesis of guaranteed stable neural network controllers with transient quality optimization. Proceedings of the TSU. Technical Sciences, 2022, no. 12, pp. 128–133. (in Russian).
  15. Abate A., Ahmed D., Giacobbe M., Peruffo A. Formal synthesis of Lyapunov neural networks. IEEE Control Systems Letters, 2021, vol. 5, no. 3, pp. 773–778.
  16. Lambora A., Gupta K., Chopra K. Genetic algorithm - a literature review. Proc. of the 2019 International Conference on Machine Learning, Big Data, Cloud and Parallel Computing (COMITCon), 2019, pp. 380–384.
  17. Katoch S., Chauhan S.S., Kumar V. A review on genetic algorithm: past, present, and future. Multimedia Tools and Applications, 2021, vol. 80, no. 5, pp. 8091–8126.
  18. Zenkin A.M., Peregudin A.A., Bobtsov A.A. Lyapunov function search method for analysis of nonlinear systems stability using genetic algorithm. arXiv, 2023, arXiv:2307.03030.
  19. Wright A. The exact schema theorem. arXiv, 2011, arXiv:1105.3538.
  20. Holland J.H. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT Press, 1992, 232 p.

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