1st Edition

# Linear Discrete-Time Systems

By

,

## Lyubomir T. Gruyitch

ISBN 9781032339382
Published June 14, 2022 by CRC Press
476 Pages

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## Book Description

This book covers crucial lacunae of the linear discrete-time time-invariant dynamical systems and introduces the reader to their treatment, while functioning under real, natural conditions, in forced regimes with arbitrary initial conditions. It provides novel theoretical tools necessary for the analysis and design of the systems operating in stated conditions. The text completely covers two well-known systems, IO and ISO, along with a new system, IIO. It discovers the concept of the full transfer function matrix F(z) in the z-complex domain, which incorporates the Z-transform of the system, input and another variable, vectors, all with arbitrary initial conditions. Consequently, it addresses the full system matrix P(z) and the full block diagram technique based on the use of F(z), which incorporates the Z-transform of the system, input and another variable, vectors, all with arbitrary initial conditions. The book explores the direct relationship between the system full transfer function matrix F(z) and the Lyapunov stability concept, definitions, and conditions, as well as with the BI stability concept, definitions, and conditions. The goal of the book is to unify the study and applications of all three classes of the linear discrete-time time-invariant system, for short systems.

Part I Basic Topics

1 Introduction

1.1 Discrete time, physical variables, and systems

1.2 Discrete time and system dynamics

1.3 Discrete-time systems and complex domain

1.4 Notational preliminaries

2 Classes of discrete-time linear systems

2.1 IO systems

2.2 ISO systems

2.3 IIO systems

2.4 System forms

3 System regimes

3.1 System regime meaning

3.2 System regimes and initial conditions

3.3 Forced and free regimes

3.4 Desired regime

3.5 Deviations and mathematical models

3.6 Stationary and nonstationary regimes

3.7 Equilibrium regime

4 Transfer function matrix G(z)

Part II Full Transfer Function Matrix F(z) And System Realization

5 Problem statement

6 Nondegenerate matrices

7 Definition of F(z)

7.1 Definition of F(z) in general

7.2 Definition of F(z) of the IO system

7.3 Definition of F(z) of the ISO system

7.4 Definition of F(z) of the IIO system

8 Determination of F(z)

8.1 F(z) of the IO system

8.2 F(z) of the ISO system

8.3 F(z) of the IIO system

8.4 Conclusion: Common general form of F(z)

9 Full block diagram algebra

9.1 Introduction

9.2 Parallel connection

9.3 Connection in series

9.4 Feedback connection

10 Physical meaning of F(z)

10.1 The IO system

10.2 The ISO system

10.3 The IIO system

11 System matrix and equivalence

11.1 System matrix of the IO system

11.2 System matrix of the ISO System

11.3 System matrix of the IIO system

12 Realizations of F(z)

12.1 Dynamical and least dimension of a system

12.2 On realization and minimal realization

12.3 Realizations of F(z) of IO systems

12.4 Realizations of F(z) of ISO systems

12.5 Realizations of F(z) of IIO systems

Part III Stability Study

13 Lyapunov stability

13.1 Lyapunov stability concept

13.2 Definitions

13.3 Lyapunov method and theorems

13.4 Conditions via F(z)

14 Bounded Input stability

14.1 BI stability and initial conditions

14.2 Definitions

14.3 Conditions

15 Motivation for the book

16 Summary of the contributions

17 Future teaching and research

V Bibliography

VI Appendices

A Notation

A.1 Abbreviations

A.2 Indexes

A.3 Letters

A.4 Names

A.5 Symbols and vectors

A.6 Units

B Z-transforms and unit impulses

B.1 Z-transforms

B.2 Unit impulses

Part VII Index

Author Index

Subject Index

...

## Author(s)

### Biography

Zoran M. Buchevats is a Certified Mechanical Engineer (Dipl. M. Eng.), Master of Mechanical Engineering Sciences (M. M. E. Sc.), and Doctor of Technical Sciences (D. Sc.) (all with the University of Belgrade-UB, Serbia). Dr Buchevats has rich academic work and experience at many Universities in Serbia (Belgrade, Kragujevac, Nis) and in Republic of Srbska (Banja Luka) and international cooperation (USA, Russia). He founded and has led the research Laboratory of Discrete Digital Control Systems within Division for Automatic Control at Faculty of Mechanical Engineering-FME, UB. He published 3 textbooks, 1 laboratory practicum, and 1 handbook (all in Serbian); 2 lecture notes (in English for foreign students); 35 scientific papers in scientific journals and conference proceedings (papers in full); led or took part in 12 projects; carried out 1 visiting presentation by invitation (USA), carried out 1 year visiting scholar (ECED U. of Wisconsin-Madison, USA), and carried out 1 study visit by invitation (Control Institute, Moscow, Russia); holds 1 patent in Serbia and holds 1 international patent application announcement and publishing. In 2013, Dr Buchevats received the annual award for the best invention for the year of 2012 by Belgrade Chamber of Commerce. He was awarded by FME in Kraljevo, UKragujevac with the Plaque (the highest rank award) in recognition and gratitude for the many years of successful cooperation and contribution to the development of the Faculty.

Lyubomir T. Gruyitch is Certified Mechanical Engineer (Dipl. M. Eng.), Master of Electrical Engineering Sciences (M. E. E. Sc.), and Doctor of Engineering Sciences (D. Sc.) (all with the University of Belgrade -UB, Serbia). Dr. Gruyitch was a leading contributor to the creation of the research Laboratory of Automatic Control, Mechatronics, Manufacturing Engineering and Systems Engineering of the National School of Engineers (Belfort, France), and a founder of the educational division and research Laboratory of Automatic Control of the Faculty of Mechanical Engineering, UB. He has given invited university seminars in Belgium, Canada, England, France, Russia, Serbia, Tunis, and USA. He has published 9 books (8 in English, 1 in Serb), 4 textbooks (in Serbo-Croatian), 11 lecture notes (7 in French, 2 in English, 2 in Serbo-Croatian), one manual of solved problems, one book translation from Russian, chapters in eight scientific books, 130 scientific papers in scientific journals, 173 conference research papers, and 2 educational papers. France honored him Doctor Honoris Causa (DHC).