J. V. Tsyganova¹, D. V. Galushkina², A. N. Kuvshinova³

¹Innopolis University (Innopolis, Russian Federation)

²Ulyanovsk State University (Ulyanovsk, Russian Federation)

³Ulyanovsk State University of Education (Ulyanovsk, Russian Federation)

Abstract. The paper proposes a new square-root gradient-based parameter identification method for discrete-time linear stochastic state-space systems with unknown input signals. A new algorithm is developed for calculating the values of the identification criterion and its gradient. The approach is based on a square-root modification of the Gillijns – De Moor method and uses numerically stable matrix orthogonal transformations. Unlike the existing solutions, this paper uses original methods for differentiating matrix orthogonal transformations. A new sensitivity model is constructed and theoretically justified, that allows calculating the values of the identification criterion gradient by using partial derivatives of state vector estimates based on identified parameters. The main results include new equations for the square-root sensitivity model and a square-root algorithm for calculating the values of the identification criterion and its gradient. Numerical experiments were performed in MATLAB for example of solving the numerical identification problem of a stochastic diffusion model with unknown boundary conditions. The effectiveness of the proposed algorithm is confirmed by comparison of gradient-based and gradient-free methods. The results of numerical experiments demonstrate efficiency of approach proposed which can be used to solve practical problems of identifying the parameters of mathematical models represented by discrete-time state-space linear stochastic systems, in the absence of any prior information about the input signals.

Key Words: parameter identification, discrete-time linear stochastic systems with unknown input signals, simultaneous input signal and state estimation, square-root filtering algorithm, filter sensitivity equations

For citation: J. V. Tsyganova, D. V. Galushkina, A. N. Kuvshinova. A square-root method for identifying the parameters of discrete-time linear stochastic systems with unknown input signals. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:3(2025), 341–363. DOI: https://doi.org/10.15507/2079-6900.27.202503.341-363

Information about the authors:

Julia V. Tsyganova, D. Sc. (Phys. and Math.), Professor, Institute of Data Science and Artificial intelligence, Innopolis University, (1, Universitetskaya Str., Innopolis, 420500, Russia), ORCID:http://orcid.org/0000-0001-8812-6035, tsyganovajv@gmail.com

Darya V. Galushkina, Assistant Professor, Department of Information Technologies, Ulyanovsk State University, (42 Leo Tolstoy str., Ulyanovsk, 432017, Russia), ORCID: https://orcid.org/0000-0003-4041-0533, dgalushkina73@gmail.com

Anastasia N. Kuvshinova, Ph.D. (Phys. and Math.) Department of Higher Mathematics, Ulyanovsk State Pedagogical University, (4/5 Lenin Square, Ulyanovsk, 432071, Russia), ORCID:http://orcid.org/0000-0002-3496-5981, kuvanulspu@yandex.ru