Course detail

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 and to give methods of their applications. Therefore the attention is focused on application examples and their utilization for study of stability of processes, their controllability and observability.

The ability to orientate in the basic notions and problems
of discrete and difference equations.
Solving problems in the areas cited in the annotation above
by use of these methods. Solving these problems by use of
modern mathematical software.

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

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 bodě "Syllabus" minutely. The discipline is recommended for Ph.D. programme students, who apply discrete and difference relations, equations and numerical algorithms. As illustration we point to mathematicas modelling of phenomena in nanotechnologies, control theory and signal processing.

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

Abilities leading to successful solution of some classes of differential equations as well as necessary theoretical knowledge will be positively estimated.

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.

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.

Saber, Elaydi, N., An Introduction to Difference Equations, Springer-Verlag, New York, Inc., 1996.
Diblík, J., Růžičková, I., Discrete Processes in Electrical Engineering, Studijní modul, Brno, 2005.