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

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 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 (integration, variable separation, linear differential equations, exact differntial equation,…).

2. Application of ODE1 and their solution in Matlab environment (Thermal exchange, Newton's laws of motion, electric circuits, ...).

3. 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.

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

5. Application of ODE of higher orders in the theory of continuous control (use of basic mathematical methods, transfer function and Matlab).

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

7. LT in transfer theory. Impulse and step functions. LT and transfer function in Matlab.

8. Application of LT in ODE. Inverse LT using the residue theorem. Laplace transform of an impulse.

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-transform. Search for impulse and step functions as a solution of difference equation and other applications of difference equations in control theory.

12. Numerical integration and differentiation. Numerical methods of solving ODE (Euler method, Runge-Kutta method, ...).

13. Presentation of particular tasks (differential equations) applied in physics, mechanics, economics, biology, etc. It will include at least 3 methods of solution (including solutions in Matlab) and a conclusion focused on the effectiveness of the methods.

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. 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.

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

5. Calculation of higher order ODE using Matlab.

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

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

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

9. Calculation of the inverse LT from the definition (using the residue theorem). Laplace transform of impulse. Assignment of credit examples.

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

11. Use of Z - transform 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.

12. Numerical solution (numerical integration and derivation, numerical methods of solving ODR).

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.