over systemen, theorie¸n en...
TRANSCRIPT
![Page 1: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/1.jpg)
Over SYSTEMEN,
THEORIEËN en THEOREMA’S
About SYSTEMS, THEORIES, and THEOREMS
Afscheidscollege, Groningen, 13 januari 2004
– p.1/26
![Page 2: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/2.jpg)
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
![Page 3: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/3.jpg)
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
![Page 4: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/4.jpg)
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
![Page 5: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/5.jpg)
The BEHAVIOR
EVENTSYSTEM
Which event sequences are possible?
– p.3/26
![Page 6: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/6.jpg)
The BEHAVIOR
SYSTEM
time
time
time
time
– p.3/26
![Page 7: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/7.jpg)
The BEHAVIOR
The behavior =
all trajectories of the system variableswhich, according to the mathematicalmodel, are possible.
– p.3/26
![Page 8: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/8.jpg)
The BEHAVIOR
time
event space
Totality of ‘legal’ trajectories =: the behavior
– p.3/26
![Page 9: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/9.jpg)
INPUT/OUTPUT SYSTEMS
SYSTEMstimulus response
causeinput
effectoutput
SYSTEM
– p.4/26
![Page 10: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/10.jpg)
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)– p.5/26
![Page 11: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/11.jpg)
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)– p.5/26
![Page 12: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/12.jpg)
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)
Norbert Wiener (1894-1964) – p.5/26
![Page 13: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/13.jpg)
The HISTORY of I/O SYSTEMS
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)
Norbert Wiener (1894-1964)Rudolf Kalman (1930- )
– p.5/26
![Page 14: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/14.jpg)
MIT, CAMBRIDGE
– p.6/26
![Page 15: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/15.jpg)
MIT, CAMBRIDGE
– p.6/26
![Page 16: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/16.jpg)
LYAPUNOV FUNCTIONS
Aleksandr Lyapunov (1857-1918)
trajectory
Lyapunovfunction
system
– p.7/26
![Page 17: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/17.jpg)
LYAPUNOV FUNCTIONS
trajectory
Lyapunovfunction
system
– p.7/26
![Page 18: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/18.jpg)
DISSIPATIVE SYSTEMS
A system is said to be dissipative �Initial storage + Supply Final storage
supplySYSTEM
– p.8/26
![Page 19: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/19.jpg)
DISSIPATIVE SYSTEMS
– p.8/26
![Page 20: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/20.jpg)
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
– p.9/26
![Page 21: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/21.jpg)
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
![Page 22: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/22.jpg)
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
– p.9/26
![Page 23: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/23.jpg)
CAMBRIDGE, MIT, GRONINGEN
– p.10/26
![Page 24: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/24.jpg)
CAMBRIDGE, MIT, GRONINGEN
– p.10/26
![Page 25: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/25.jpg)
CAMBRIDGE, MIT, GRONINGEN
– p.10/26
![Page 26: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/26.jpg)
AUTONOMOUS SYSTEMS
PAST
time
FUTURE
event space
time
event space
Autonomous := past implies future
– p.11/26
![Page 27: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/27.jpg)
PLANETARY MOTION
Planet ???
How can it move?
– p.12/26
![Page 28: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/28.jpg)
PLANETARY MOTION
Johannes Kepler (1571-1630)
– p.12/26
![Page 29: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/29.jpg)
PLANETARY MOTION
Kepler’s first law
PLANET
SUN Ellipse, sun in focus
– p.12/26
![Page 30: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/30.jpg)
PLANETARY MOTION
Kepler’s second law
DC
B
A= areas in = times
– p.12/26
![Page 31: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/31.jpg)
PLANETARY MOTION
Kepler’s third law
1 year
34 months
(period)
�
= (diameter)
�
– p.12/26
![Page 32: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/32.jpg)
Troja, Hiroshima, Srebrenica,
Maar de planeet zij draaide voort!
– p.13/26
![Page 33: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/33.jpg)
The STATE of the PLANET
position velocity
The state = position and velocity
– p.14/26
![Page 34: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/34.jpg)
The EQUATION of the PLANET
acceleration = function of position and velocity
�� �
� � � � � � � �� �� � � �
– p.15/26
![Page 35: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/35.jpg)
The EQUATION of the PLANET
acceleration = function of position and velocity
�� �
� � � � � � � �� �� � � �
� � �
��
�
– p.15/26
![Page 36: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/36.jpg)
Isaac Newton (1643-1727)
Hypotheses non
fingo
– p.16/26
![Page 37: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/37.jpg)
Isaac Newton (1643-1727)
Hypotheses non
fingo
– p.16/26
![Page 38: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/38.jpg)
NEWTON’s LAWS
Newton’s 2nd law: force = mass * acceleration
’
w2−nd law
F’
– p.17/26
![Page 39: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/39.jpg)
NEWTON’s LAWS
The law of gravitation: attraction =mass
(distance)
�
F’’
wGRAVITY
� � �
��
�
– p.17/26
![Page 40: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/40.jpg)
NEWTON’s LAWS
Newton’s third law: the sum of the forces = 0
��
’
��������
F’’
2 GRAVITY
w
F’
– p.17/26
![Page 41: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/41.jpg)
NEWTON’s LAWS
� � � � � � ��� � � � �
� � � � � �
��� �� �
!!� " # � $ �
� � � � � � � � � �� � �
��
�
– p.17/26
![Page 42: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/42.jpg)
The HISTORY of CLOSED SYSTEMS
Henri Poincare (1854-1912)
George Birkhoff (1884-1944)
Stephen Smale (1930- )– p.18/26
![Page 43: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/43.jpg)
A system as a behavior
A fully general, consistent state construction
– p.19/26
![Page 44: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/44.jpg)
The BRACHISTOCHRONE
Johann Bernoulli (1667-1748)– p.20/26
![Page 45: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/45.jpg)
The BRACHISTOCHRONE
B
A
Let the ball roll as fast as possible from to !
– p.20/26
![Page 46: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/46.jpg)
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
– p.21/26
![Page 47: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/47.jpg)
The BRACHISTOCHRONE
time
???
???
time
h = verticaldisplacement
x = horizontaldisplacement
What is the behavior?– p.22/26
![Page 48: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/48.jpg)
The BRACHISTOCHRONE
B
Adisplacementx = horizontal
h = verticaldisplacement
conservation of energy
vertical displacement = (velocity)
�
� � ���
� � � ��� ' � �
Behavioral equation, defines the behavior.– p.22/26
![Page 49: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/49.jpg)
Prima la Musica,
poi le Parole.
Prima la Fısica,
poi la Matematica.
– p.23/26
![Page 50: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/50.jpg)
Prima la Musica,
poi le Parole.
Prima la Fısica,
poi la Matematica.
– p.23/26
![Page 51: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/51.jpg)
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
![Page 52: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/52.jpg)
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
![Page 53: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/53.jpg)
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
![Page 54: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/54.jpg)
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
![Page 55: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/55.jpg)
THANKS
– p.25/26
![Page 56: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/56.jpg)
THANKS
– p.25/26
![Page 57: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/57.jpg)
THANKS
Charles Desoer (1926- )
– p.25/26
![Page 58: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/58.jpg)
THANKS
Charles Desoer (1926- )
George Zames (1934-1997)– p.25/26
![Page 59: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/59.jpg)
THANKS
Charles Desoer (1926- )
George Zames (1934-1997)
Rudolf Kalman (1930- )
– p.25/26
![Page 60: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/60.jpg)
THANKS
Charles Desoer (1926- )
George Zames (1934-1997)
Rudolf Kalman (1930- ) Roger Brockett (1938- )
– p.25/26
![Page 61: Over SYSTEMEN, THEORIE¸N en THEOREMA’Shomes.esat.kuleuven.be/~jwillems/Lectures/2004/Lastslides.pdfOver SYSTEMEN, THEORIE¸N en THEOREMA’S About SYSTEMS, THEORIES, and THEOREMS](https://reader030.vdocuments.mx/reader030/viewer/2022040707/5e0782ba19c6e471f01b895a/html5/thumbnails/61.jpg)
Thank youThank you
Thank you
Thank you
Thank you
Thank you
Thank you
Thank you– p.26/26