Differential and Difference Equations in Control Theory (FSI-VDR-K)

The aim of this course is to apply ordinary differential and difference equations in control theory. Furthermore, the subject is an effort to expand and connect knowledge in the field of solving differential and difference equations, Laplace transform, Z-transform and transfer theory. The purpose of the course will also be to solve and simulate tasks with the support of the Matlab program. 

By completing this course, students will not only deepen their knowledge in the field of differential and difference equations, but they will get acquainted with applications and various solutions, including their advantages and disadvantages (classical mathematical approach, Laplace transform, Z-transform, Matlab). 

Differential and integral calculus of a function of one variable, Differential equations, Difference equations, Linear continuous and discrete control, Matlab. 

The course is focused on the deepening and application of the theory of differential and difference equations in the theory of regulation. In this course, emphasis is placed on specific applications of these equations in the theory of continuous and discrete control, including their demonstrations in the Matlab environment. The content of the course is the use of Laplace and Z-transform. For clarity, the tasks will be solved and simulated in Matlab. 

The course is taught in the form of closely related exercises and lectures. Lectures have the character of an explanation of basic theory and methods. The exercise is focused on the practical mastery of the topics covered in lectures, including computer support and simulation in the Matlab environment. 

The condition for awording  assessment is active participation in exercises and elaboration of the assigned example (resp. students can choose their own example), on which the student demonstrates different methods (including computer processing) and evaluates their effectiveness. 

The examination is written and oral. In the written part the student solves two basic topics (differential and difference equations). The oral part of the exam contains a discussion of these tasks and possible supplementary questions. 

Attendance at seminars and lectures is mandatory, due to the close interconnection of their content. Absences can be compensated by assigning substitute tasks. 

1. Introduction (Motivation). Ordinary differential equations of the 1st order (ODE1). Basic concepts. Methods of solution of ODE1(variable separation, linear differential equations, exact differntial equation,…).

2. Application of ODE1 and their solution in Matlab environment.

3. Introduction of Laplace transform (LT). Basic concepts. Calculation of direct LT from definition. Basic LT sentences and operator dictionary.

4. LT in transfer theory. Impulse and transient functions. LT and transmission in Matlab.

5. Application of LT in ODE1.

6. ODE of higher orders. Construction of solution of a homogeneous n-th order linear differential equation. The method of undetermined coefficients for nonhomogeneous linear differential equation of the n-th  order.

7. ODE of higher orders (equations of motion, course of oscillations of electric current in RCL circuit).

8. Calculation of higher order ODE using basic mathematical approaches, transfer theory and Matlab.

9. Difference of a variable, difference equation with positive and negative displacements.

10. Methods of solving difference equations (classical method, numerical method).

11. Z-transfer, solution of difference equations using Z-transfer. Search for impulse and step functions as a solution of difference equation and other applications of difference equations in control theory.

12. Addendum to LT: Calculation of direct LT from definition. Inverse LT using the residue theorem. Laplace transformation of an impulse.

13. Final summary of the course. 

The exercise is closely related to the content of lectures:

1. Basic methods of solution of ordinary differential equations of the first order (ODE1), including their interpretation in Matlab environment.

2. Application of  ODE1 in the theory of linear control.

3. Use of direct and inverse Laplace transform (LT). Repetition of the partial fraction decomposition

4. LT in transfer theory. Impulse and step functions.

5. Application of LT in ODE1. LT and transfer theory in Matlab.

6. ODE of higher orders. Solution of homogeneous linear differential equation (LDE) of the n-th order. Indefinite coefficient method for nonhomogeneous n-th order LDE.

7. Calculation of higher order ODE using basic mathematical methods.

8. Calculation of higher order ODE using transfer theory and Matlab.

9. Solution of difference equations with positive and negative displacements in numerical approach ('open solution') and classical approach.

10. Use of Z - transfer in solving difference equations. Search for impulse and step functions as solutions of difference equations and other applications of difference equations in the theory of discrete control.

11. Assignment of credit examples. Addendum to LT: Calculation of direct LT from definition. Inverse LT using the residue theorem. Laplace transformation of impulse.

12. Presentation of credit (assessment) examples.

13. Presentation of credit examples (assessment).  Assessments.

Švarc, I., Matoušek, R., Šeda, M., Vítečková, M.: Automatizace-Automatické řízení, skriptum VUT FSI v Brně, CERM 2011.