Course detail

Algebraic Theory of Control

FSI-VTRAcad. year: 2023/2024

The students will be provided with the principles of the algebraic theory of discrete linear control. The basic algebraic concepts and methods used in the theory will be discussed. The main interest will be focused on the study of polynomials, because they are the
most important tools of the theory of discrete linear control. First, the fundamentals of the theory of rings and the theory of formal series will be expounded. This will be followed by the study of polynomials (as special cases of formal series) and polynomial matrices from the view-point of the theory of discrete linear control. This will be done with the help of the fundamental knowledge of the theory of rings.

Language of instruction


Number of ECTS credits


Mode of study

Not applicable.

Entry knowledge

The knowledge of mathematics gained within the bachelor's study programme.

Rules for evaluation and completion of the course

The graded course-unit credit is awarded on condition of having passed a written test at the end of the semester.
Since the attendance at lectures is not compulsory, it will not be checked, and compensation of possible absence will not be required.


The goal of the course is to acquaint students with the mathematical principles that form the basis of the algebraic theory of discrete linear control and that are used for solving problems of the theory.
Students will be made familiar with solving mathematical problems that occur in the theory of discrete linear control. Basic problems of this kind concern the synthesis of optimal control, which is reduced to searching for solutions of linear polynomial equations (as the transmission of a system can be expressed by using polynomials).

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

V. Kučera: Algebraic Theory of Discrete-Time Linear Control. Academia, Praha 1978. (EN)
Kučera V.: Discrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester 1979. (EN)
Grimble M. J., Kučera V.: Polynomial Methods for Control Systems Design. Springer, London 1996. (EN)

Recommended reading

V.Kučera: Algebraická teorie diskrétního lineárního řízení, Academia, Praha, 1978 (CS)
J.Karásek, J.Šlapal: Teorie okruhů pro diskrétní lineární řízení, FSI VUT v Brně, 2000 (učební text) (CS)
J.Karásek, J.Šlapal: Polynomy a zobecněné polynomy v teorii řízení, Akademické nakladatelství CERM, Brno, 2007 (CS)
Paul M. Cohn, Introduction to Ring Theory, Springer Undergraduate Mathematics Series, 2000, pp. 229 (EN)


Classification of course in study plans

  • Programme N-AIŘ-P Master's, 2. year of study, summer semester, compulsory
  • Programme N-MAI-P Master's, 2. year of study, summer semester, compulsory-optional

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Introduction
2.-3. Rings
4.-5. Fields
6.-7. Formal power series
8.-9. Polynomials
10.-11. Polynomial fractions
12.-13. Polynomial matrices