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Project titleSolving the problems of parameter identification of discrete stochastic systems with additive and multiplicative noises: new methods and array algorithms based on matrix orthogonal transformations

AffiliationFederal State Budgetary Educational Institution of Higher Education "Ulyanovsk State Pedagogical University",

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-220 - Mathematical simulation of technical systems

Keywordsdiscrete stochastic systems, parameter identification, multiplicative noises, orthogonal transformations, array algorithms

Annotation
Discrete linear stochastic systems with additive and multiplicative noises describe a wide class of mathematical models of complex systems in many fields of science and technology, such as complex production and technological systems, energy, technical and economic systems, image processing systems, telecommunication systems, aerospace systems, etc. During the functioning of the system, the structure of the mathematical model may remain unchanged, however, the parameters of the model may vary. As a result, there is a need to solve the problem of parameter identification, which in practice is usually solved by numerical methods. In software implementation of algorithms, a serious problem is the influence of machine round-off errors on the computation result. Therefore, the construction of numerically stable methods is an extremely important and urgent problem. One of the effective solutions to this problem is to use numerically stable matrix orthogonal transformations. Currently, a wide class of numerically stable array algorithms has been developed for both optimal discrete filtering and adaptive filtering and parameter identification. But these methods are constructed only for the class of discrete linear stochastic systems with additive Gaussian noise. Such methods have not yet been developed for systems with additive and multiplicative noise. The scientific novelty of the project lies in the fact that for the first time it posed and will solve the problem of constructing new, insensitive to machine rounding errors, parameter identification methods in the class of discrete stochastic systems with additive and multiplicative noises. The main result of the project is the creation of a holistic research tool in the form of mathematical methods, algorithms, and software for solving the problems of parameter identification of discrete stochastic systems with additive and multiplicative noises. The expected results will make a significant contribution to the development of the theory of numerically stable methods of adaptive filtering and parameter identification. The constructed array algorithms will make it possible to obtain high-quality solutions to the problems of parameter identification of mathematical models represented by discrete linear stochastic systems with additive and multiplicative noises.

Expected results
Expected results of the project: 1. New algorithms for information filtering for a class of discrete stochastic systems with multiplicative and additive noise. 2. New array square-root discrete filtering algorithms (in covariance and informational form) based on the matrix QR transform for a class of discrete stochastic systems with multiplicative and additive noise. 3. New array algorithms for discrete filtering (in covariance and informational form) based on modified weighted Gram-Schmidt orthogonalization (MWGS-transform) for a class of discrete stochastic systems with multiplicative and additive noise. 4. Adaptive filters based on the developed array algorithms with the ability to calculate the sensitivity values from the uncertainty parameters of a discrete stochastic system with multiplicative and additive noise. 5. Algorithms for parameter identification of discrete stochastic systems with additive and multiplicative noise based on array orthogonal transformations. 6. Library of functions with the implementation of all algorithms in MATLAB. The scientific significance of the results: The results of the project implementation will make a significant contribution to the development of the theory of numerically stable discrete filtering methods for stochastic systems with additive and multiplicative noises. The constructed array algorithms will make it possible to obtain high-quality solutions to the problems of parameter identification of mathematical models represented by equations in the state space. Applied significance of the results: The results of the project implementation can be used to solve the problems of parametric identification of models in many areas of science and technology, such as complex production and technological, energy, technical and economic systems, image processing systems, telecommunication systems, aerospace systems, etc.

Annotation of the results obtained in 2023
All work on the project during the reporting period was carried out in accordance with the scientific research plan. At the second stage of the project, the following main results were obtained: 1. New parameter identification algorithms have been developed for systems with additive and multiplicative noises based on gradient-free numerical optimization methods, in which the values of the identification criterion are calculated using array discrete filtering algorithms. 2. New parameter identification algorithms have been developed for systems with additive and multiplicative noise based on gradient-type numerical optimization methods, in which the values of the identification criterion and its gradient are calculated using array adaptive discrete filtering algorithms with the ability to calculate sensitivity values of the state vector estimates with respect to uncertainty parameters. 3. Software implementation of the developed algorithms in MATLAB language was completed. The results obtained during the second stage of the project are new and can be used in solving problems of processing measurement information and identification for discrete linear stochastic systems with multiplicative and additive noises. To obtain the planned results, the following work was performed: 1. When constructing new gradient-free parameter identification algorithms, an approach was used based on minimizing the identification criterion in the form of a negative log-likelihood function, depending on the measurement residuals calculated by the standard Kalman filter. Expressions were obtained for calculating the identification criterion in terms of quantities calculated by array filtering algorithms (SVD-based and UD-based filters in covariance form were considered). To minimize the obtained identification criteria, gradient-free metaheuristic minimization algorithms were used, such as genetic algorithm and simulated annealing method. Computational experiments have been carried out to demonstrate the advantages of new algorithms in terms of resistance to machine rounding errors compared to conventional algorithms. 2. When constructing new gradient-type parameter identification algorithms, an approach was used based on minimizing the identification criterion in the form of a negative log-likelihood function, but with the construction of array adaptive filters, in which, in addition to state vector estimates, partial derivatives of state vector estimates with respect to uncertainty parameters are calculated. These estimates are used to calculate the gradient of the identification criterion, which allows the use of various gradient optimization algorithms to minimize the identification criterion. During the implementation of the second stage of the project, methods for calculating derivatives in matrix orthogonal transformations for systems with additive noises were generalized for the case of systems with additive and multiplicative noises. As an example, an adaptive covariance UD-based filter was considered. Computational experiments have been carried out to demonstrate the advantages of gradient identification algorithms in identification speed compared to gradient-free algorithms. 3. During the implementation of the second stage of the project, functions and scripts were added to the library of discrete filtering algorithms for discrete-time linear stochastic systems with additive and multiplicative noises, developed at the first stage, for solving parameter identification problems: - identification criterion based on the negative log-likelihood function; - SVD-based identification criterion; - UD-based identification criterion; - UD-based identification criterion with gradient calculation; - functions for minimizing identification criteria; - auxiliary functions and scripts. The results obtained during the second stage of the project were presented in the form of oral presentations at the following international scientific conferences: 1) IX International Conference and Youth School “Information Technologies and Nanotechnologies” (ITNT-2023), April 17–23, 2023, Samara, Russia. 2) IX International Scientific and Practical Conference “Control Systems, Complex Systems: Modeling, Stability, Stabilization, Intelligent Technologies” (CSMSSIT-2023), April 24–25, 2023, Yelets, Russia. The following articles have been prepared and published: 1) Tsyganov, A.V. Identification of parameters of the discrete-time stochastic systems models with multiplicative and additive noises / A.V. Tsyganov, Yu. V. Tsyganova, A.V. Golubkov // Information technologies and nanotechnologies (ITNT-2023): proceedings of the IX International Conference and Youth School (Samara, April 17–23, 2023). — T.5. Data Science. — Samara: Samara University Publishing House, 2023. — P.050682. (In Russian) 2) Kuvshinova, A.N. On evaluation of derivatives in a gradient-based parameter identification algorithm for discrete-time stochastic systems with multiplicative and additive noises / A.N. Kuvshinova, A.V. Tsyganov, Yu.V. Tsyganova // Control systems, complex systems: modeling, stability, stabilization, intelligent technologies: materials of the IX International Scientific and Practical Conference, Yelets, April 24–25, 2023. — Yelets: Yelets State University named after. I. A. Bunina, 2023. — P. 79–84. (In Russian) 3) Tsyganov, A. SVD-based identification of parameters of the discrete-time stochastic systems models with multiplicative and additive noises using metaheuristic optimization / A. Tsyganov, Y. Tsyganova // Mathematics. — 2023. — Vol.11, no.20. — URL: https://www.mdpi.com/2227-7390/11/20/4292. — DOI: 10.3390/math11204292 (WoS Q1, Scopus Q2) 4) Tsyganov, A. Parameter Identification of the Discrete-Time Stochastic Systems with Multiplicative and Additive Noises Using the UD-based State Sensitivity Evaluation / A. Tsyganov, Y. Tsyganova // Mathematics. — 2023. (WoS Q1, Scopus Q2, accepted) During the second stage of the project, a computer program was registered: Tsyganov, A.V. Library of discrete filtering and parameter identification algorithms for discrete-time linear stochastic systems with additive and multiplicative noises v1.0 / A.V. Tsyganov, Yu.V. Tsyganova, A.V. Golubkov // ROSPATENT. Certificate of state registration of a computer program No. 2023686525, 12/07/2023. A report on the progress of the project and the results obtained is available on the website of the Laboratory of Mathematical Modeling of Ulyanovsk State Pedagogical University named after. I.N. Ulyanov: http://lmm.ulspu.ru/ru/content/rscf_22-21-00387

Publications

1. Kuvshinova A.N., Tsyganov A.V., Tsyganova Y.V. О вычислении производных в алгоритме параметрической идентификации градиентного типа для дискретных стохастических систем с мультипликативными и аддитивными шумами Системы управления, сложные системы: моделирование, устойчивость, стабилизация, интеллектуальные технологии: материалы IX Международной научно-практической конференции, Елец, 24–25 апреля 2023 года, С. 79–84 (year - 2023)

2. Tsyganov A.V., Tsyganova Y.V. SVD-based identification of parameters of the discrete-time stochastic systems models with multiplicative and additive noises using metaheuristic optimization Mathematics, Vol. 11, 4292 (year - 2023) https://doi.org/10.3390/math11204292

3. Tsyganov A.V., Tsyganova Y.V. Parameter Identification of the Discrete-Time Stochastic Systems with Multiplicative and Additive Noises Using the UD-based State Sensitivity Evaluation Mathematics, - (year - 2023)

4. Tsyganov A.V., Tsyganova Y.V., Golubkov A.V. Идентификация параметров моделей дискретных стохастических систем с мультипликативными и аддитивными шумами Информационные технологии и нанотехнологии (ИТНТ-2023): сборник трудов по материалам IX Международной конференции и молодежной школы (г. Самара, 17–23 апреля 2023 г.), Т. 5. Науки о данных. — С. 050682. (year - 2023)

5. - Библиотека алгоритмов дискретной фильтрации и параметрической идентификации для дискретных линейных стохастических систем с аддитивными и мультипликативными шумами v1.0 -, 2023686525 (year - )

Annotation of the results obtained in 2022
In the course of the first phase of the project, all the planned work has been completed in full and the following main results have been obtained: 1. New algorithms of information filtering for a class of discrete stochastic systems with multiplicative and additive noises were developed and theoretically justified. 2. New block-orthogonalized square-root algorithms of discrete filtering (in covariance and information form) based on array QR-transformation for a class of discrete stochastic systems with multiplicative and additive noises were developed and theoretically proved. 3. New block-orthogonalized algorithms of discrete filtering (in covariance and information form) based on the modified weighted Gram-Schmidt orthogonalization (MWGS-transformation) for a class of discrete stochastic systems with multiplicative and additive noises were developed and theoretically proved. 4. The software implementation of the developed algorithms in MATLAB language was performed. The obtained results were presented at the VIII International Conference and Youth School "Information Technologies and Nanotechnology" (ITNT-2022), Samara, Russia, 23-27 May, 2022 (Samara National Research University, Institute for Image Processing Systems RAS - branch of Research Center "Crystallography and Photonics" RAS). An oral report "New discrete-time filtering algorithms based on the MWGS-orthogonalization for systems with multiplicative and additive noises" was presented in the section "Data Science" and was awarded the Best Paper Award (III place). Based on the results of the first stage, the following articles were prepared for publication and published: 1. Tsyganov, A. V. Extended square-root covariance filtering algorithm for discrete-time systems with multiplicative and additive noises / A. V. Tsyganov, Yu. V. Tsyganova, T. N. Kureneva // Lobachevskii Journal of Mathematics. - 2022. - Vol. 43, no. 6. - P. 1438-1445. - DOI: 10.1134/S199508022209027X. 2. New algorithms of discrete filtering based on MWGS-orthogonalization for systems with multiplicative and additive noise / A. V. Tsyganov, Yu. - Vol. 5 Data sciences. - Samara: Samara University Publisher, 2022. - С. 051332. Discrete filtering algorithms based on modified weighted Gram-Schmidt orthogonalization for discrete stochastic systems with multiplicative and additive noises / A. V. Tsyganov, Yu. (accepted). The report on the progress of the project and the results obtained is available on the website of the Laboratory of Mathematical Modeling of the Ulyanovsk State Pedagogical University named after Ulyanov: http://lmm.ulspu.ru/ru/content/rscf_22-21-00387 Works on the second stage of the project were started. The paper "Model Parameters Identification of Discrete Stochastic Systems with Multiplicative and Additive Noise" was prepared for presentation at the IX International Conference and Youth School "Information Technologies and Nanotechnology" (ITNT-2023), Samara, Russia, April 17-21, 2023 (under review).

Publications

1. Tsyganov A.V., Tsyganova Yu.V., Kureneva T.N. Extended Square-Root Covariance Filtering Algorithm for Discrete-Time Systems with Multiplicative and Additive Noises Lobachevskii Journal of Mathematics, Vol. 43, No. 6, pp. 1200–1207 (year - 2022) https://doi.org/10.1134/S199508022209027X

2. Tsyganov A.V., Tsyganova Yu.V., Kuvshinova A.N., Golubkov A.V. Алгоритмы дискретной фильтрации на основе модифицированной взвешенной ортогонализации Грама-Шмидта для дискретных стохастических систем с мультипликативными и аддитивными шумами Вычислительные технологии, Т. 28, № 5. С. 70–86. (year - 2023) https://doi.org/10.25743/ICT.2023.28.5.007

3. Tsyganov A.V., Tsyganova Yu.V., Kuvshinova A.N., Kureneva T.N. Новые алгоритмы дискретной фильтрации на основе MWGS-ортогонализации для систем с мультипликативными и аддитивными шумами Информационные технологии и нанотехнологии (ИТНТ-2022): сборник трудов по материалам VIII Международной конференции и молодежной школы (г. Самара, 23–27 мая)., Т. 5. Науки о данных. — С. 051332. (year - 2022)

4. - В УлГПУ завершился первый этап работ по проекту Российского научного фонда под руководством профессора кафедры высшей математики А.В. Цыганова Улпресса, Дата публикации: 09.12.2022. Время публикации: 14:28 (year - )