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Saint-Petersburg University
Faculty of Applied Mathematics – Control Processes
Evgeny I. Veremey, Professor
Margarita Sotnikova, Senior Lecturer, PhD
Digital Systems of Control
and Signal Processing
Introduction and Course Content
General Preface
CBU Winter School in Information Technology
Saint Petersburg, 27 January - 5 February 2011
Introduction to digital systems
of control and signal processing
Course 1. "Digital systems of control and signal processing"
Prof. E. Veremey, Senior Lecturer M. Sotnikova, SPbU, Russia
Course 2. “Computer graphics and image processing"
Assoc. Prof. I. Makeev, SPbU, Russia
Course 2. “Image Compression"
Assist. Prof. A. Kolesnikov, University of Eastern Finland
Peter the Great's ukase (edict)
regarding the foundation of a
university in St. Petersburg
was approved by the Senate of
the Russian Empire on 28
January 1724. A characteristic
feature of this, Russia's first,
institution of higher education
was its organic link with the
other two constituents of St.
Petersburg's scholastic trinity:
the Academy of Sciences and
the Grammar School.
University today:
Today St Petersburg State University is a major Russian center of science, education
and culture of international repute. Many important universities in Europe, America, and
Asia - notably Cambridge University (UK), Bologna University (Italy), the Free University
of Berlin and Hamburg University (Germany), Carlton University (Canada), Amsterdam
University (the Netherlands), Stockholm University (Sweden), the Municipal University
of Osaka (Japan), to name but a few, - have maintained various academic contacts with
the University of St. Petersburg.
The discoveries and achievements of the University professors and graduates have
become part of the history of both international and Russian science and technology.
Eight of them were awarded Nobel Prizes - I.Pavlov (1904) and I.Mechnikov (1908) for
physiology and medicine; N.Semenov (1956) for chemistry; L.Landau (1962) and
A.Prokhorov (1964) for physics; V.Leontyev (1973) and L.Kantorovich (1975) for
economics, I.Brodsky (1985) for literature.
21 faculties - 11 Scientific Institutes - Academicals School over 25.000 students - near 2.000 Postgraduate students - near 5.000 lecturers &
researchers - over 40 members of Russian Academies of Science, Education etc.
AMPC Faculty of SPbU
The Faculty of Applied Mathematics and Control Processes is Russia's first
faculty in this field. It was opened in 1969 on the several departments base of
Mathematics and Mechanics Faculty, as well as a few laboratories belonging to
the Research Institute of Mathematics and Mechanics of Leningrad University.
In their teaching and research the staff draw on the rich traditions of the St
Petersburg-Leningrad mathematics school, which is connected with such worldfamous names as L.Euler, P.Chebyshev, A.Liapunov, A.Markov, and others.
The Faculty is a major training and research center capable to solve any
scientific problems involving the use of computer-based technologies and the
training or retraining of professionals in this field.
The Faculty graduates are fully qualified to work successfully in any sphere that
requires the use of applied mathematics and computer science. Today, this
sphere extends to practically every kind of human activity - from defense and
high tech down to biology, management, ecology, finance, economics, and
communications.
1500 students – 100 post graduators (PhD students) – 250 educational
and research staff (40 Doctors of science, 90 PhD) – 20 departments – 9
scientific laboratories – 5 teaching laboratories
AMPC Faculty of SPbU
Educational directions for graduators:
Applied Mathematics & Informatics
Applied Mathematics & Physics,
Informational Technologies
Educational programs:
4-year bachelor studying
2-year master studying
5-year specialist studying
Educational directions for post graduators (PhD students):
- Automation of Production & Technological Processes
- Automation & Software for Computers, Computer Complexes,
Systems & Networks
- Computational Mathematics
- Differential Equations
- Mathematical Cybernetics
- Mechanics of Deformed Solids
- Systems for Automated Design
- Computers Application, Mathematical Modeling & Methods in Research
AMPC Faculty of SPbU
Educational activities:
The faculty is engaged in fundamental research. The results are regularly published in
the leading national and international scientific journals, are reported on the
international workshops, conferences and congresses. The list of scientific books
published by the Faculty staff members exceeds 500 titles. Starting from 1969 more
than 500 Ph. D. and about 100 Dr. Sci. dissertations were prepared and defended.
Many different scientific enterprises are under way.
The applications selected for study frequently have nonstandard features that, in turn,
leads to creation of new mathematical methods and tools. The area of particular
interest has always been the control theory with its various applications.
The research and educational programs are administered through the structure of
departments and laboratories. A number of seminars support the research activities.
It has become a tradition of the faculty to carry out regular weekly seminars in the
Theory of Control, Game Theory, Numerical Analysis, Problems of the Deformable
Solid Mechanics and others.
The educational process is based on the best traditions of St.Petersburg mathematical
school, inspired by such well-known mathematicians as L.Euler, P.L.Chebyshev,
A.M.Lyapunov, A.A.Markov and many others.
AMPC Faculty of SPbU
Curriculum (general obligatory courses):
Mathematical Analysis
Higher Algebra
Geometry
Complex Analysis
Differential and Integral Equations
Calculus of Variations
Mathematical Programming
Mathematical Physics
Numerical Analysis
Control Theory
Game Theory and Operations Research
Probability Theory and Mathematical Statistics
Mechanics of the Controlled Motion
Deformed Solid Mechanics
Computers and Programming Languages
Informational Logical Systems
Control Models
Some Photos
of our everyday
life
Content of the Course
1
INTRODUCTION
Part 1. MATHEMATICAL MODELS of DIGITAL SYSTEMS
1. Linear transformation of digital signals
2. Mathematical Models of DLTI systems on the base of difference equations
3. Z-transformation (Laurent transformation)
4. Mathematical models of DLTI systems in z-domain
Part 2. ANALISYS OF DIGITAL SIGNALS and SYSTEMS
1. The Discrete Time Fourier series and Transformation
2. The Discrete Fourier Transformation
3. Frequency responses of DLTI systems
4. Stability of DLTI systems
5. Performance Indices of Digital Systems
1
Content of the Course
Part 3. SYNTHESIS of DIGITAL SYSTEMS
1. Оptimization approach to digital systems analysis and design
2. Parametrical optimization with given admissible dynamical region
3. Modal approach to the synthesis
4. Optimal LQR synthesis
Cart fixed by the spring
2
d2y
dy
m 2  p  ky  0 (1)
dt
dt
y0
1
0.5
dy
0
dt
0
-0.5
-1
0
2
4
6
y(t )  et c1 cos t  c2 sin t  (2)
8
10
Controlled cart
3
y(t )
- output of sensor
d2y
dy
m 2  p  ky  F (t )  u (3)
dt
dt
Feedback :
Methods of control
1. u   F (t )
а) by disturbance
2. u  ky
б) by output
3. u  u(t )
Program Control
Controlled cart
3
Coding of
Analog Signal
1.5
1
0.5
а) initial analog signal
0
-0.5
-1
-1.5
1.5
1
0.5
б) result of filtering
0
-0.5
-1
-1.5
1.5
1
0.5
в) digital sygnal
0
-0.5
-1
-1.5
0
5
10
15
t(s)
20
25
30
Digital Processing of Analog Signal
6
y (t )
Preliminary Analog
Processing
(Analog Filter)
~
y (t )
Processing Scheme
Coding
Analog-to-Digital
Conversion
(А/D)
y[n]
Digital
Transformation
(Digital Processor)
Digital Processing
u[n]
Digital-to-Analog
Conversion
(D/A)
~ (t )
u
Smoothing of
Processed Signal
(Analog Filter)
u(t )
Encoding
Controlled cart
7
Encoding of
Digital Signal
1.5
1
0.5
a) digital signal
0
-0.5
-1
-1.5
1.5
1
0.5
b) result of D/A
0
-0.5
-1
-1.5
1.5
1
0.5
c) result of smoothing
0
-0.5
-1
-1.5
0
5
10
15
t(s)
20
25
30