Over SYSTEMEN,

THEORIEËN en THEOREMA’S

About SYSTEMS, THEORIES, and THEOREMS

Afscheidscollege, Groningen, 13 januari 2004

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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

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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

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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

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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

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INPUT/OUTPUT SYSTEMS

SYSTEMstimulus response

causeinput

effectoutput

SYSTEM

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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?

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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

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