The specifics of teaching mathematics in technical universities

D. N. Bogacheva

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2024), pp. 21-29

Résumé

This paper discusses the methodology of explanation the mathematical tools for studying control theory, which allows not only to present the necessary material to the students, but also to explain the essence of the obtained solutions, as well as the methods of obtaining them. The presentation of the material is considered using the example of the controlled object trajectory constructing problem. The methodology consists of a step-by-step presentation of the necessary material. First, a brief statement of the problem is formulated, allowing to understand how this problem arises, what initial data are needed for its formulation, and how the solution looks like (using the example of constructing the object trajectory by solving the Cauchy problem). Then the formal definition of the main mathematical tool used to solve the problem, its characteristics and description are considered (in the example there are formal definition of the fundamental matrix and its main properties explained with a meaningful definition). Based on the described tool, formal solution to the problem and its interpretation are given (that is the fixation of the formal solution). At this stage, the basic theoretical information necessary for the student to use the described tool is formed (using the example of obtaining a generalized equation of object motion in Cauchy form using a fundamental matrix, as well as the equation of motion for a system with constant parameters). Next, a direct solution to the simplest version of the stated problem is presented (for a first-order system), based on the use of elementary mathematical knowledge (direct solution of a differential equation by the method of separation of variables), which allows to understand the physics of the solution process (that is the fixation of the meaningful solution). The example shows two components of motion: the first one is free, depending on the initial conditions, and the second one is forced, depending on the input of the system. Finally, a comparison is made between the formal solution and the meaningful solution of the simplest version of the stated problem, and the connection between these two solutions is analyzed. The example compares the free and forced components of the formal and direct solutions. One of the options for finding the fundamental matrix using a power series is also shown. In conclusion, recommendations are given for the development of lectures and accompanying practical classes, based on the proposed methodology.

Détail

Keywords: trajectory, control object, linearization, Cauchy problem, fundamental matrix, matrix exponential.