I.N. Sinitsyn1, V.I. Sinitsyn2, E.R. Korepanov3, T.D. Konashenkova4

1−4 Federal Research Center «Computer Science and Control» of the Russian Academy of Sciences (Moscow, Russia)
1 sinitsin@dol.ru, 2 vsinitsin@frccsc.ru, 3 ekorepanov@frccsc.ru, 4 tkonashenkova64@mail.ru

Paper is devoted to approximate analytical synthesis by mean square criterion (MSC) of one dimensional nonlinear nonstationary stochastic system with high availability (StSHA) on the basis of statistical linearization method (SLM), theory of multiscale analysis (MSA), wavelet canonical expansions (WLCE) and wavelet neural network (WNN). Observable StSHA is described by Pugachev nonlinear discontinuous output and input stochastic processes (StP). Output and input StP depend on vector of Gaussian random parameters (RP). Output and output StP contain Gaussian StP and additive Gaussian noise being independent from RP. SLM is used for RP linearization of nonlinear functions. Linear operator of MSC optimal StSHA is approximated by means MSA and WNN. Input StP is presented as linear combination of input random variables (RV) by WLCE. MSC estimate is combination of operator equation. Formulae for accuracy of MSC estimates are given. Experimental software tools are developed. Computer experiments confirm of approximate analytical mean square estimates in comparison with statistical estimates.

Sinitsyn I.N., Sinitsyn V.I., Korepanov E.R., Konashenkova T.D. Approximate analytical synthesis of nonlinear stochastic systems with high availability by mean square criterion // Highly Available Systems. 2026. V. 22. № 2. P. 18−32. DOI: https://doi.org/10.18127/
j20729472-202602-02

1. Pugachyov V.S. Teoriya sluchajny`x funkcij i eyo primenenie k zadacham avtomaticheskogo upravleniya. M.: Fizmatgiz.1962. 884 s.
2. Sinicyn I.N. Kanonicheskie predstavleniya sluchajny`x funkcij. Teoriya i primeneniya. 2-e izd. M.: Torus Press. 2023. 816 s.
3. Zhu B., Karimi H.R., Zhang L., Zhao X. Neural network-based adaptive reinforcement learning for optimized backstepping tracking control of nonlinear systems with input delay. Applied Intelligence. 2025. V. 55. № 129. https://doi.org/10.1007/s10489-024-05932-x
4. Musavi N., Sun D., Mitra S., Dullerud G.E., Shakkottai S. Verification and Parameter Synthesis in Stochastic Systems with Hybrid State Space Using Optimistic Optimization. IEEE Open Journal of Control Systems. 2023. V. 2. P. 263–276. https://doi.org/10.1109/ojcsys.2023.3299152
5. Chen Z., Ma W. A Bayesian approach to data-driven multi-stage stochastic optimization. Journal of Global Optimization. 2024. V. 90.
   P. 401–428. https://doi.org/10.1007/s10898-024-01410-3
6. Diveev A., Konstantinov S. Applying Neural Networks for the Identification of Control Object Mathematical Models for the Control Problems. 8th International Conference on Control, Decision and Information Technologies (CoDIT). IEEE. 2022. P. 1059–1063.
7. Xajkin S. Nejronny`e seti: polny`j kurs. Per. s angl. 2-e izd. SPb.: Dialektika. 2020. 1104 s.
8. Pugachyov V.S. Lekcii po funkcional`nomu analizu. M.: Izd. MAI. 1996. 744 s.
9. Dobeshi I. Desyat` lekcij po vejvletam. M.; Izhevsk: NICz «Regulyarnaya i xaoticheskaya dinamika». 2004. 464 s.
10. Terexov S.A. Vejvlety` i nejronny`e seti. Nauchnaya sessiya MIFI – 2001: III Vseros. nauchn.-texn. konf. «Nejroinformatika-2001»: lekcii po nejroinformatike. M.: MIFI. 2001. S. 142–181.
11. Veitch D. Wavelet Neural Networks and their application in the study of dynamical system. Networks, 2005. V. 1. № 8. P. 313–320.
12. Sinicyn I.N., Sinicyn V.I., Korepanov E`.R., Konashenkova T.D. Modelirovanie nestacionarnogo stoxasticheskogo processa posredstvom ego kanonicheskogo razlozheniya na osnove vejvlet-nejronnoj seti. Sistemy` i sredstva informatiki, 2024. T. 34. № 2. S. 21–39. doi: 10.14357/08696527240202
13. Sinitsyn I., Sinitsyn V., Korepanov E., Konashenkova T. Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Cano­nical Expansions. Mathematics. 2022. V. 10. Iss. 9. Art. 1517. 14 p. https://doi.org/10.3390/math10091517
14. Sinicyn I.N., Sinicyn V.I., Korepanov E`.R., Konashenkova T.D. Nejrosetevoj algoritm sinteza optimal`noj linejnoj stoxasticheskoj sistemy` vy`sokoj dostupnosti po e`nergeticheskomu kriteriyu. Sistemy` vy`sokoj dostupnosti. 2024. T. 20. № 4. S. 5–14. doi: 10.18127/j20729472-202404-01 (in Russian)