Course detail

# Discrete Methods in Civil Engineering 2

The discipline is devoted to description of processes via discrete equations. It consists of three parts:
a) Stability of solutions. Stability of numerical algorithms.
b) Application of difference equations.
c) Control of processes using difference equations.

Language of instruction

Czech

Number of ECTS credits

10

Mode of study

Not applicable.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

The ability to orientate in the basic notions and problems
of discrete and difference equations.
Solving problems in the areas cited in the annotation.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Stability of equilibrium points. Kinds of stabily and instability.
2. Stability of linear systems with the variable matrix.
3. Stability of nonlinear systems via linearization.
4. Ljapunov direct method of stability.
5. Phase analysis of two-dimensional linear discrete system with constant matrix, classification of equilibrium points.
6. Application of difference equations. Multiple-room heating problem. Newton law of cooling.
7. Discrete equivalents of continuous systems.
8. Discrete control theory.
9. The controllability and the complete controllability.
10. Matrix of controllability, the canonical forms of controllability, controllable canonical form, construction of the control algorithm.
11. Observability, complete observability, nononservability, principle of duality, the observability matrix, canonical forms of observability, relation of controllability and observability.
12.–13. Stabilization of control by feedback.

Work placements

Not applicable.

Aims

The purpose of continuation of this course is analysis of stability of linear and non-linear discrete systems and methods of their applications.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

J. Diblík. Diskrétní metody ve stavebnictví I, studijní materiál, 82 stran (CS)
Diblík. Diskrétní metody ve stavebnictví II, studijní materiál, 66 stran (EN)
Elaydi, Saber N. An Introduction to Diﬀerence Equations, Third Edition, Springer, 2005  (EN)
Michael A. Radin. Difference Equations For Scientists And Engineering: Interdisciplinary Difference Equations, ‎ World Scientific, 2019 (EN)

Farlow, S.J. An Introduction to Diﬀerential Equations, Dover Publications, 2006 (EN)
Lakshmikantham, V., Trigiante, Donato. Theory of Diﬀerence Equations, Numerical Methods and Applications, Second Edition, Marcel Dekker, 2002 (EN)

Classification of course in study plans

• Programme DPA-S Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPC-S Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-S Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-S Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPC-E Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-E Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPA-E Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-E Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-GK Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPA-GK Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPC-GK Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-GK Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-M Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPA-M Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPC-M Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-M Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPA-K Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPC-K Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-K Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-K Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKA-V Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DKC-V Doctoral, 2. year of study, winter semester, compulsory-optional
• Programme DPA-V Doctoral, 2. year of study, winter semester, compulsory-optional

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Stability of equilibrium points. Kinds of stabily and instability. 2. Stability of linear systems with the variable matrix. 3. Stability of nonlinear systems via linearization. 4. Ljapunov direct method of stability. 5. Phase analysis of two-dimensional linear discrete system with constant matrix, classification of equilibrium points. 6. Application of difference equations. Multiple-room heating problem. Newton law of cooling. 7. Discrete equivalents of continuous systems. 8. Discrete control theory. 9. The controllability and the complete controllability. 10. Matrix of controllability, the canonical forms of controllability, controllable canonical form, construction of the control algorithm. 11. Observability, complete observability, nononservability, principle of duality, the observability matrix, canonical forms of observability, relation of controllability and observability. 12.–13. Stabilization of control by feedback.