Course detail

Discrete Processes in Electrical Engineering

FEKT-DPC-MA2Acad. year: 2022/2023

The discipline is devoted to description of processes via discrete equations. It consists of three parts:
a) basic calculus and basic methods of analysis of discrete processes,
b) application of difference equations, investigation of stability processes,
c) application of difference equations in control of processes.

The plan of discipline is described in the point "Syllabus" in detail. The discipline is recommended for Ph.D. programme students, who will apply discrete and difference relations, equations and numerical algorithms also. As illustration we point to mathematical modelling of phenomena in nanotechnologies, control theory and signal processing.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

The ability to orientate in the basic notions and problems of discrete and difference equations. Solving problems in the areas cited in the curriculum by use of these methods. Solving given by use of modern mathematical software. Main outcomes are:

1) Ability to solve basic classes of difference equations of the first order.
2) Usage of difference equations of first order to solution of equations describing various phenomena modeled by difference equations of the first order. Transformation of differential equations to discrete equations, modeling electrical circuits by difference equations.
3) Finding of equilibria points of scalar equations, determination stability and other properties of solutions in the vicinity of equilibria.
4) Construction of cob-web diagrams for inverstigation of stability of equailibrium points.
5) Determination of stability of numerical algorithms using equilibrium points.
6) Application of basic formulae of discrete calculus.
7) Solution of homogeneous and non-homogeneous linear discrete equations of higher-order.
8) Construction of solutions of systems homogeneous and non-homogeneous difference equations of the first order.
9) Solution of linear homogeneous system of difference equations by Putzer algorithm. Finding of a particular solution.
10) Determination of stability and non-stability of nonlinear and linear discrete systems by method of a fundamental matrix and by Lyapunov method.
11) Application of Z-transform to solution of linear difference equations of higher-order and to solution of linear difference systems.
12) Detection of controllability and observability of linear discrete systems.

Prerequisites

The subject knowledge on the Bachelor´s and Magister´s degree level is requested from mathematical disciplines.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Abilities leading to successful solution of some typical classes of difference equations as well as necessary theoretical knowledge and its application will be positively estimated. During half-year term students must prepare 3 essay. The final evaluation (examination) depends on assigned points (0 points is minimum, 100 points is maximum), 30 points is maximum points which can be assigned for essays. Final examination is in written and oral form and is estimated
as follows: 0- points is minimum, 70 points is maximum.

Course curriculum

1. Basic notions and methods of investigation of discrete processes.
2. Discrete calculus (some difference relations based on corresponding continuous relations). Difference equations and systems.
3. Basic notions used in difference equations (equilibrium points, periodic points, eventually equilibrium points and eventually periodic points, stability of solution, repelling and attracting points) and their illustration on examples (modelling of circuits with the aid of difference equations, the transmission of information).
4. Recursive algorithms of solutions of systems of discrete equations and equations of higher order (the case of constant coefficients, the method of variation of parameters, the method of variation of constants).
5. Construction of the general solution. Transformation of some nonlinear equations into linear equations. Difference equations modelled with the aid of sampling, impulses inputs, computation of characteristic from the signal response (response of Dirac distribution), transmission effects.
6. Application of difference equations – stability of processes. Stability of equilibrium points. Kinds of stabily and instability.
7. Stability of linear systems with the variable matrix. Stability of nonlinear systems via linearization.
8. Ljapunov direct method of stability.
9. Phase analysis of two-dimensional linear discrete system with constant matrix, classification of equilibrium points.
10. Application of difference equations - control of processes. Discrete equivalents of continuous systems.
11. Discrete control theory (the controllability, the complete controllability, matrix of controllability, the canonical forms of controllability, controllable canonical form, construction of the control algorithm).
12. Observability, complete observability, nonobservability, principle of duality, the observability matrix.
13. Canonical forms of observability, relation of controllability and observability. Stabilization of control by feedback.

Work placements

Not applicable.

Aims

Discrete and difference equations are the mathematical base of many fields of engineering science. The purpose of this course is to develop the basic notions concerning the properties of solutions of such equations, demonstrate methods of their solution, give methods for investigation of stability of solutions, clarify their utilization in control theory and show their applications. Therefore the attention is focused on application examples and their utilization for study of stability of processes, their controllability and observability.

Specification of controlled education, way of implementation and compensation for absences

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year. Necessary condition (for final examination) are three prepared essays motivated by published papers with applications of discrete equations.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Diblík, J., Diskrétní metody v elektroinženýrství, elektronický text, Brno, 2014 (CS)
Mickens, Ronald E., Difference Equations: Theory, Applications and Advanced Topics, Third Edition, Chapman & Hall/CRC, 2016 (CS)
Oppenheim, Alan, V., Schaffer, Ronald, W., Discrete-Time Signal Processing, 3rd Edition, Pearson, 2014 (CS)

Recommended reading

Miček, J., Jurečka, M., Moderné prostriedky implementácie metód číslicového spracovania signálov I.. EDIS, Žilina, 2013 (CS)
Sami Fadali, M., Visioli, A., Digital Control Engineering, Analysis and Design, 2nd Edition, Elsewier, AP, 2013 (CS)

Classification of course in study plans

  • Programme DPC-EKT Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-IBE Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-KAM Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-MET Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-SEE Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-TEE Doctoral 0 year of study, summer semester, compulsory-optional
  • Programme DPC-TLI Doctoral 0 year of study, summer semester, compulsory-optional

Type of course unit

 

Seminar

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

I. Basic notions and methods of investigation of discrete processes (5 weeks). Discrete calculus
(some difference relations based on corresponding continuous relations). Difference equations and systems. Basic notions used in difference equations (equilibrium points, periodic points, eventually equilibrium points and eventually periodic points, stability of solution, repelling and attracting points) and their illustration on examples (modelling of circuits with the aid of difference equations, the transmission of information). Recursive algorithms of solutions of systems of discrete equations and equations of higher order (the case of constant coefficients, the method of variation of parameters, the method of variation of constants). The computer construction of the general solution. Transformation of some nonlinear equations into linear equations. Difference equations modelled with the aid of sampling, impulses inputs, computation of characteristic from the signal response (response of Dirac distribution), transmission effects.
II. Application of difference equations – stability of processes (4 weeks).
Stability of equilibrium points. Kinds of stabily and instability. Stability of linear systems with the variable matrix. Stability of nonlinear systems via linearization. Ljapunov direct method of stability. Phase analysis of two-dimensional linear discrete system with constant matrix, classification of equilibrium points.
III. Application of difference equations - control of processes (4 weeks).
Discrete equivalents of continuous systems. Discrete control theory (the controllability, the complete controllability, matrix of controllability, the canonical forms of controllability, controllable canonical form, construction of the control algorithm). Observability (complete observability, nononservability, principle of duality, the observability matrix, canonical forms of observability, relation of controllability and observability). Stabilization of control by feedback.