# Automated Linearization of a System of Nonlinear Ordinary Differential Equations

## Authors

• Gulzat Ziyatbekova Al-Farabi Kazakh National University, IT department, Almaty
• Aigerim Mazakova Al-Farabi Kazakh National University, Almaty
• Sholpan Jomartova Al-Farabi Kazakh National University, Almaty
• Waldemar Wójcik Lublin University of Technology
• Talgat Mazakov Al-Farabi Kazakh National University, Almaty

## Abstract

The article explores the possibility of automated linearization of nonlinear models. The process of constructing a linearized model for a nonlinear system is quite laborious and is practically unrealizable when the dimension n is greater than 3. Also, when building a linearized model, there is a "human factor" that does not guarantee the correctness of the calculation of arithmetic expressions. In this regard, the task of automating the process of linearization of the original nonlinear model is relevant. On the basis of linearized models, one can further investigate such properties of the original model as stability and controllability. Based on the application of computer algebra, a constructive algorithm for the linearization of a system of nonlinear ordinary differential equations has been developed. Developed software on MatLab. The efficiency of the proposed algorithm has been demonstrated on applied problems. For a mathematical model of the dynamics of an unmanned aerial vehicle, described by a system of ordinary differential equations of the 6th order, a linearized system is automatically constructed. Two procedures have been developed for the mathematical model of a two-link robot. The first procedure allows one to normalize the mathematical model obtained on the basis of the Lagrange equations of the second kind. The second procedure uses the normalized mathematical model to build its linearized copy.

The linearized models obtained were further used to investigate the stability of the original models.

In order to account for possible inaccuracies in the measurement of model technical parameters arising from wear and tear or variability in temperature operating conditions, the linearized model is assumed to be interval.

For the interval linearized model, the procedure of constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. Based on the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the system under study is analyzed.

Keywords— Ordinary differential equation, Computer algebra, Stability, Controllability, MatLab