Course unit code: 
FEKTDMA2A 
Academic year: 
2017/2018 
Type of course unit: 
optional specialized 
Level of course unit: 
Doctoral (3rd cycle) 
Year of study: 
1 
Semester: 
summer 
Number of ECTS credits: 
4 
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 cobweb 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 nonhomogeneous linear discrete equations of higherorder.
8) Construction of solutions of systems homogeneous and nonhomogeneous 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 nonstability of nonlinear and linear discrete systems by method of a fundamental matrix and by Lyapunov method.
11) Application of Ztransform to solution of linear difference equations of higherorder and to solution of linear difference systems.
12) Detection of controllability and observability of linear discrete systems.


Mode of delivery:
90 % facetoface, 10 % distance learning


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


Corequisites:
Not applicable.


Recommended optional programme components:
Not applicable.


Course contents (annotation):
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.


Recommended or required reading:
Saber, Elaydi, N., An Introduction to Difference Equations, SpringerVerlag, New York, Inc., third edition, 2005. Farlow, S. J., An Introduction to Differential Equations, McGrawHill, Inc., 1994 Diblík, J., Růžičková, I., Discrete Processes in Electrical Engineering, Studijní modul, Brno, 2012.


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 halfyear 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 form and is estimated
as follows: 0 points is minimum, 70 points is maximum.


Language of instruction:
English


Work placements:
Not applicable.


Course curriculum:
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 twodimensional 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, nonobservability, principle of duality, the observability matrix, canonical forms of observability, relation of controllability and observability). Stabilization of control by feedback.


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.

