MATHEMATICAL MODELING AND NUMERICAL SIMULATION

A general approach to constructing gradient methods for parameter identification based on modified weighted Gram – Schmidt orthogonalization and information-type discrete filtering algorithms

A. V. Tsyganov^a, J. V. Tsyganova<sup>abc

a</sup> Ulyanovsk State Pedagogical University, 4/5 Lenin sq., Ulyanovsk, 432071, Russia
^b Innopolis University, 1 Universitetskaya st., Innopolis, 420500, Russia
^c Ulyanovsk State University, 42 L. Tolstoy st., Ulyanovsk, 432017, Russia

Abstract: The paper considers the problem of parameter identification of discrete-time linear stochastic systems in the state space with additive and multiplicative noise. It is assumed that the state and measurements equations of a discrete-time linear stochastic system depend on an unknown parameter to be identified.
A new approach to the construction of gradient parameter identification methods in the class of discrete-time linear stochastic systems with additive and multiplicative noise is presented, based on the application of modified weighted Gram – Schmidt orthogonalization (MWGS) and the discrete-time information-type filtering algorithms.
The main theoretical results of this research include: 1) a new identification criterion in terms of an extended information filter; 2) a new algorithm for calculating derivatives with respect to an uncertainty parameter in a discrete-time linear stochastic system based on an extended information LD filter using the direct procedure of modified weighted Gram – Schmidt orthogonalization; and 3) a new method for calculating the gradient of identification criteria using a “differentiated” extended information LD filter.
The advantages of this approach are that it uses MWGS orthogonalization which is numerically stable against machine roundoff errors, and it forms the basis of all the developed methods and algorithms. The information LD-filter maintains the symmetry and positive definiteness of the information matrices. The algorithms have an array structure that is convenient for computer implementation.
All the developed algorithms were implemented in MATLAB. A series of numerical experiments were carried out. The results obtained demonstrated the operability of the proposed approach, using the example of solving the problem of parameter identification for a mathematical model of a complex mechanical system.
The results can be used to develop methods for identifying parameters in mathematical models that are represented in state space by discrete-time linear stochastic systems with additive and multiplicative noise.

Keywords: parameter identification, gradient-based method, MWGS orthogonalization, algorithm of information filtering, linear discrete-time stochastic system, parameter uncertainty

UDC: 681.5.015

Received: 27.06.2025
Revised: 01.10.2025
Accepted: 07.10.2025

DOI: 10.20537/2076-7633-2025-17-5-761-782