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Over SYSTEMEN,
THEORIEËN en THEOREMA’S
About SYSTEMS, THEORIES, and THEOREMS
Afscheidscollege, Groningen, 13 januari 2004
– p.1/26
OPEN DYNAMICAL SYSTEMS
‘system’ := the object which we are studying
‘dynamical’ :=
we are interested in its evolution over time
‘open’ :=
the system interacts with its environment
– p.2/26
OPEN DYNAMICAL SYSTEMS
‘system’ := the object which we are studying
‘dynamical’ :=
we are interested in its evolution over time
‘open’ :=
the system interacts with its environment
– p.2/26
OPEN DYNAMICAL SYSTEMS
‘system’ := the object which we are studying
‘dynamical’ :=
we are interested in its evolution over time
‘open’ :=
the system interacts with its environment
– p.2/26
The BEHAVIOR
EVENTSYSTEM
Which event sequences are possible?
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The BEHAVIOR
SYSTEM
time
time
time
time
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The BEHAVIOR
The behavior =
all trajectories of the system variableswhich, according to the mathematicalmodel, are possible.
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The BEHAVIOR
time
event space
Totality of ‘legal’ trajectories =: the behavior
– p.3/26
INPUT/OUTPUT SYSTEMS
SYSTEMstimulus response
causeinput
effectoutput
SYSTEM
– p.4/26
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)– p.5/26
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)– p.5/26
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)
Norbert Wiener (1894-1964) – p.5/26
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)
Norbert Wiener (1894-1964)Rudolf Kalman (1930- )
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MIT, CAMBRIDGE
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MIT, CAMBRIDGE
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LYAPUNOV FUNCTIONS
Aleksandr Lyapunov (1857-1918)
trajectory
Lyapunovfunction
system
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LYAPUNOV FUNCTIONS
trajectory
Lyapunovfunction
system
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DISSIPATIVE SYSTEMS
A system is said to be dissipative �Initial storage + Supply Final storage
supplySYSTEM
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DISSIPATIVE SYSTEMS
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To COUNT or to THINK?
105
80
1973
9085
number of citations
2000
039575
year
40
20
Years Citations
1973-1982 36
1983-1992 31
1993-now 256
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To COUNT or to THINK?
impact factor for year x
citations in year x to articles publishedin the journal in years x-1 and x-2
=number of articles published in the journal
in years x-1 and x-2
105
80
1973
9085
number of citations
2000
039575
year
40
20
Years Citations
1973-1982 36
1983-1992 31
1993-now 256
– p.9/26
To COUNT or to THINK?
105
80
1973
9085
number of citations
2000
039575
year
40
20
Years Citations
1973-1982 36
1983-1992 31
1993-now 256
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CAMBRIDGE, MIT, GRONINGEN
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CAMBRIDGE, MIT, GRONINGEN
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CAMBRIDGE, MIT, GRONINGEN
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AUTONOMOUS SYSTEMS
PAST
time
FUTURE
event space
time
event space
Autonomous := past implies future
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PLANETARY MOTION
Planet ???
How can it move?
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PLANETARY MOTION
Johannes Kepler (1571-1630)
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PLANETARY MOTION
Kepler’s first law
PLANET
SUN Ellipse, sun in focus
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PLANETARY MOTION
Kepler’s second law
DC
B
A= areas in = times
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PLANETARY MOTION
Kepler’s third law
1 year
34 months
(period)
�
= (diameter)
�
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Troja, Hiroshima, Srebrenica,
Maar de planeet zij draaide voort!
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The STATE of the PLANET
position velocity
The state = position and velocity
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The EQUATION of the PLANET
acceleration = function of position and velocity
�� �
� � � � � � � �� �� � � �
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The EQUATION of the PLANET
acceleration = function of position and velocity
�� �
� � � � � � � �� �� � � �
� � �
��
�
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Isaac Newton (1643-1727)
Hypotheses non
fingo
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Isaac Newton (1643-1727)
Hypotheses non
fingo
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NEWTON’s LAWS
Newton’s 2nd law: force = mass * acceleration
’
w2−nd law
F’
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NEWTON’s LAWS
The law of gravitation: attraction =mass
(distance)
�
F’’
wGRAVITY
� � �
��
�
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NEWTON’s LAWS
Newton’s third law: the sum of the forces = 0
��
’
��������
F’’
2 GRAVITY
w
F’
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NEWTON’s LAWS
� � � � � � ��� � � � �
� � � � � �
��� �� �
!!� " # � $ �
� � � � � � � � � �� � �
��
�
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The HISTORY of CLOSED SYSTEMS
Henri Poincare (1854-1912)
George Birkhoff (1884-1944)
Stephen Smale (1930- )– p.18/26
A system as a behavior
A fully general, consistent state construction
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The BRACHISTOCHRONE
Johann Bernoulli (1667-1748)– p.20/26
The BRACHISTOCHRONE
B
A
Let the ball roll as fast as possible from to !
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SYSTEMS
+
I
C
RC
L
L
V− R
system
environment
voltage, current
voltage, current
force, position, torque, angle
force, position
% % % %% % % %% % % %& & && & && & & , 22fp
1f, 1 p
10
0 1
.
I
I(x,t)
V(x,t)Vx+
−
+
−V I
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The BRACHISTOCHRONE
time
???
???
time
h = verticaldisplacement
x = horizontaldisplacement
What is the behavior?– p.22/26
The BRACHISTOCHRONE
B
Adisplacementx = horizontal
h = verticaldisplacement
conservation of energy
vertical displacement = (velocity)
�
� � ���
� � � ��� ' � �
Behavioral equation, defines the behavior.– p.22/26
Prima la Musica,
poi le Parole.
Prima la Fısica,
poi la Matematica.
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Prima la Musica,
poi le Parole.
Prima la Fısica,
poi la Matematica.
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RECAPITULATION
A dynamical system = a behavior
From Kepler (closed system)to Newton’s second law (open system)
Dissipative systems
Almost invariant subspaces,disturbance decoupling,
pole placement by static feedback,instability by disc encirclement,
multipliers,
Mais ou sont les neiges d’antan?
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RECAPITULATION
A dynamical system = a behavior
From Kepler (closed system)to Newton’s second law (open system)
Dissipative systems
Almost invariant subspaces,disturbance decoupling,
pole placement by static feedback,instability by disc encirclement,
multipliers,
Mais ou sont les neiges d’antan?
– p.24/26
RECAPITULATION
A dynamical system = a behavior
From Kepler (closed system)to Newton’s second law (open system)
Dissipative systems
Almost invariant subspaces,disturbance decoupling,
pole placement by static feedback,instability by disc encirclement,
multipliers,
Mais ou sont les neiges d’antan?
– p.24/26
RECAPITULATION
A dynamical system = a behavior
From Kepler (closed system)to Newton’s second law (open system)
Dissipative systems
Almost invariant subspaces,disturbance decoupling,
pole placement by static feedback,instability by disc encirclement,
multipliers, ( ( (
Mais ou sont les neiges d’antan?– p.24/26
THANKS
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THANKS
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THANKS
Charles Desoer (1926- )
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THANKS
Charles Desoer (1926- )
George Zames (1934-1997)– p.25/26
THANKS
Charles Desoer (1926- )
George Zames (1934-1997)
Rudolf Kalman (1930- )
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THANKS
Charles Desoer (1926- )
George Zames (1934-1997)
Rudolf Kalman (1930- ) Roger Brockett (1938- )
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Thank youThank you
Thank you
Thank you
Thank you
Thank you
Thank you
Thank you– p.26/26