SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven,...
Transcript of SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven,...
![Page 1: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/1.jpg)
SYSTEMS THEORYand
PHYSICAL MODELS
Jan C. WillemsK.U. Leuven, Belgium
Universidade de Aveiro, 3 de junho 2004
– p.1/40
![Page 2: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/2.jpg)
Problematique
Develop a suitable mathematical frameworkfor discussing dynamical systems
aimed at modeling, analysis, and synthesis.
control, signal processing, system id., � � �
engineering systems, economics, physics, � � �
– p.2/40
![Page 3: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/3.jpg)
Paradigmatic examples - thermodynamics
Work(Q ,T )hh
W
terminal
(Q ,T )cc
terminalCooling
Heatingterminal
ThermodynamicEngine
– p.3/40
![Page 4: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/4.jpg)
Paradigmatic examples - thermodynamics
Work(Q ,T )hh
W
terminal
(Q ,T )cc
terminalCooling
Heatingterminal
ThermodynamicEngine
Express the first and second law ?
– p.3/40
![Page 5: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/5.jpg)
Paradigmatic examples - modeling
¡ Model this interconnected system !
Tearing, zooming & linking
– p.4/40
![Page 6: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/6.jpg)
Paradigmatic examples - modeling
Tearing:
– p.4/40
![Page 7: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/7.jpg)
Paradigmatic examples - modeling
Tearing:
– p.4/40
![Page 8: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/8.jpg)
Paradigmatic examples - modeling
Zooming:
– p.4/40
![Page 9: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/9.jpg)
Paradigmatic examples - modeling
Linking:
– p.4/40
![Page 10: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/10.jpg)
Paradigmatic examples - modeling
Linking:
– p.4/40
![Page 11: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/11.jpg)
THEMES
1. Closed systems
2. Input/output ;input/state/output systems
3. Beyond causality: behavioral systems
Submodule thm
Elimination thm
Controllability and image representation thm
– p.5/40
![Page 12: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/12.jpg)
THEMES
1. Closed systems
2. Input/output ;input/state/output systems
3. Beyond causality: behavioral systems
Submodule thm
Elimination thm
Controllability and image representation thm
– p.5/40
![Page 13: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/13.jpg)
How it all began ...
– p.6/40
![Page 14: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/14.jpg)
Planet ???
How does it move?
– p.7/40
![Page 15: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/15.jpg)
Kepler’s laws
Johannes Kepler (1571-1630)
SUN
PLANET
Kepler’s first law:
Ellipse, sun in focus– p.8/40
![Page 16: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/16.jpg)
Kepler’s laws
Johannes Kepler (1571-1630) DC
B
A
Kepler’s second law:
= areas in = times– p.8/40
![Page 17: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/17.jpg)
Kepler’s laws
Johannes Kepler (1571-1630)34 months
1 year
Kepler’s third law:
(period)� ��� (diameter)
�
– p.8/40
![Page 18: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/18.jpg)
The state of the planet
What determines the orbit uniquely?
SUN
PLANET
– p.9/40
![Page 19: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/19.jpg)
The state of the planet
What determines the orbit uniquely?
SUN
PLANET
The position?– p.9/40
![Page 20: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/20.jpg)
The state of the planet
What determines the orbit uniquely?
SUN
PLANET
The position and the direction of motion?– p.9/40
![Page 21: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/21.jpg)
The state of the planet
position velocity
The state = the position & the velocity
– p.9/40
![Page 22: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/22.jpg)
The equation of the planet
Consequence:acceleration = function of position and velocity
� ��� � � � � � �
��� � � �
via calculus and calculation
– p.10/40
![Page 23: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/23.jpg)
The equation of the planet
Consequence:acceleration = function of position and velocity
� ��� � � � � � �
��� � � �
via calculus and calculation
� ��� �
�� ��� �
� � � � �
– p.10/40
![Page 24: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/24.jpg)
Newton’s laws
� � � � � ��� � � �(2-nd law)
� � � � � ��� �� ��! " � # � � (gravity)
� � � � � � � � $(3-rd law)
� �� � �
� � ��� �
� � � � �
– p.11/40
![Page 25: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/25.jpg)
Kepler’s lawsK.1, K.2, & K.3
Hypotheses non
fingo
– p.12/40
![Page 26: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/26.jpg)
Kepler’s lawsK.1, K.2, & K.3
� ��� � � � ��� �� ��! " � # � � � $
Hypotheses non
fingo
– p.12/40
![Page 27: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/27.jpg)
Kepler’s lawsK.1, K.2, & K.3
� ��� � � � ��� �� ��! " � # � � � $
Hypotheses non
fingo
– p.12/40
![Page 28: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/28.jpg)
Kepler’s lawsK.1, K.2, & K.3
� ��� � � � ��� �� ��! " � # � � � $
Hypotheses non
fingo
� � ��� �
– p.12/40
![Page 29: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/29.jpg)
The paradigm of closed systems
– p.13/40
![Page 30: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/30.jpg)
‘Axiomatization’
K.1, K.2, & K.3
‘dynamical systems’, flows
closed systems as paradigm of dynamics
– p.14/40
![Page 31: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/31.jpg)
‘Axiomatization’
K.1, K.2, & K.3
% �% � � &(' ) * " � #
+ ��� � ��� � + � �
‘dynamical systems’, flows
closed systems as paradigm of dynamics
– p.14/40
![Page 32: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/32.jpg)
‘Axiomatization’
K.1, K.2, & K.3
% �% � � &(' ) * " � #
+ ��� � ��� � + � �
%% � , � & , )
‘dynamical systems’, flows
closed systems as paradigm of dynamics
– p.14/40
![Page 33: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/33.jpg)
‘Axiomatization’
K.1, K.2, & K.3
% �% � � &(' ) * " � #
+ ��� � ��� � + � �
%% � , � & , )
‘dynamical systems’, flows
closed systems as paradigm of dynamics
– p.14/40
![Page 34: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/34.jpg)
‘Axiomatization’
K.1, K.2, & K.3
% �% � � &(' ) * " � #
+ ��� � ��� � + � �
%% � , � & , )
‘dynamical systems’, flows
closed systems as paradigm of dynamics– p.14/40
![Page 35: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/35.jpg)
‘Axiomatization’
Henri Poincare (1854-1912)
George Birkhoff (1884-1944)
Stephen Smale (1930- )
– p.15/40
![Page 36: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/36.jpg)
‘Axiomatization’
XA dynamical system is defined by
a state space anda state transition function- . . . such that . . .
� / �
= state at time�
starting from state /
SYSTEM
ENVIRONMENT
Boundary
a property of but just as much of
– p.16/40
![Page 37: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/37.jpg)
‘Axiomatization’
XA dynamical system is defined by
a state space anda state transition function- . . . such that . . .
� / �
= state at time�
starting from state /
How could they forget about Newton’s second law,about Maxwell’s eq’ns, about thermodynamics,
about tearing & zooming & linking, ...?
SYSTEM
ENVIRONMENT
Boundary
a property of but just as much of
– p.16/40
![Page 38: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/38.jpg)
‘Axiomatization’
Reply: assume ‘fixed boundary conditions’
SYSTEM
ENVIRONMENT
Boundary
an absurd situation: to model a system,we have to model also the environment!
a property of but just as much of
– p.16/40
![Page 39: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/39.jpg)
‘Axiomatization’
SYSTEM
ENVIRONMENT
Boundary
Chaos theory, cellular automata, sync, etc.,‘function’ in this framework ...
a property of but just as much of
– p.16/40
![Page 40: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/40.jpg)
‘Axiomatization’
SYSTEM
ENVIRONMENT
Boundary
Chaos: not a property of the physical laws,but just as much of what the system is
interconnected to.
– p.16/40
![Page 41: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/41.jpg)
‘Axiomatization’
SYSTEM
ENVIRONMENT
Boundary
Turbulence may not be a property ofNavier-Stokes, but just as much of
the boundary conditions.
– p.16/40
![Page 42: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/42.jpg)
Meanwhile, in engineering, ...
– p.17/40
![Page 43: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/43.jpg)
Input/output systems
SYSTEMstimulus response
causeinput
effectoutput
SYSTEM outputinput
– p.18/40
![Page 44: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/44.jpg)
The originators
Lord Rayleigh (1842-1919)
Oliver Heaviside (1850-1925)
Norbert Wiener (1894-1964)
– p.19/40
![Page 45: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/45.jpg)
Mathematical description
SYSTEM outputinput
0 12 354 6 78
or 9 : ; 1 2 < 2 = 3> 1 2 = 3? 2 =
0 1 2 34 ; 8 1 2 3A@ 79 : ;CB 12 < 2 = 3> 1 2 = 3 ? 2 =@
79 :
7 D9 : ;FE 12 < 2 =G 2 = < 2 = = 3> 1 2 = 3> 1 2 = = 3 ? 2 =? 2 = =@ H H H
These models fail to deal with ‘initial conditions’.
A physical system is SELDOMLY an i/o map
– p.20/40
![Page 46: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/46.jpg)
Mathematical description
SYSTEM outputinput
0 12 354 6 78
or 9 : ; 1 2 < 2 = 3> 1 2 = 3? 2 =
These models fail to deal with ‘initial conditions’.
A physical system is SELDOMLY an i/o map
– p.20/40
![Page 47: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/47.jpg)
Input/state/output systems
%% � , � & ,I ) I � & ,I )
Rudolf Kalman (1930- ) – p.21/40
![Page 48: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/48.jpg)
‘Axiomatization’
State transition function:&(' I JI )K
state at time
�from L using input M.
X
Read-out function:& JI N ) Koutput with state O and input value P.
– p.22/40
![Page 49: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/49.jpg)
The input/state/output view turned out to bea very effective and fruitful paradigm
for control (stabilization, robustness, ...)
outputs
Actuators SensorsPlant
Controllerinputscontrol outputsmeasured
inputsto−be−controlledexogenous
prediction of one signal from another
system ID: models from data
understanding system representations(state, transfer f’n, etc.)
etc., etc., etc.
– p.23/40
![Page 50: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/50.jpg)
The input/state/output view turned out to bea very effective and fruitful paradigm
for control (stabilization, robustness, ...)
prediction of one signal from another
system ID: models from data
understanding system representations(state, transfer f’n, etc.)
etc., etc., etc.
– p.23/40
![Page 51: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/51.jpg)
Let’s take a closer look at the i/o framework ...
in control (only)
– p.24/40
![Page 52: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/52.jpg)
Difficulties with i/o
active control
outputs
Actuators SensorsPlant
Controllerinputscontrol outputsmeasured
inputsto−be−controlledexogenous
versus passive control Dampers, heat fins, pressurevalves, ...
Controlling turbulence
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 53: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/53.jpg)
Difficulties with i/o
active control
outputs
Actuators SensorsPlant
Controllerinputscontrol outputsmeasured
inputsto−be−controlledexogenous
versus passive control
Controlling turbulence
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 54: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/54.jpg)
Difficulties with i/o
active control versus passive control
Controlling turbulence
for airplanes, sharks, dolphins, golf balls, bicyclinghelmets, etc.
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 55: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/55.jpg)
Difficulties with i/o
active control versus passive control
Controlling turbulence
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 56: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/56.jpg)
Difficulties with i/o
active control versus passive control
Controlling turbulence
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 57: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/57.jpg)
Difficulties with i/o
active control versus passive control
Controlling turbulence
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 58: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/58.jpg)
Difficulties with i/o
active control versus passive control
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 59: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/59.jpg)
Difficulties with i/o
active control versus passive control
Nagano 1998
These are beautiful controllers!
But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 60: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/60.jpg)
Difficulties with i/o
active control versus passive control
Nagano 1998
These are beautiful controllers! But, the only peoplenot calling this ”control”, are the control engineers ...
– p.25/40
![Page 61: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/61.jpg)
At last, a consistent framework ...
– p.26/40
![Page 62: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/62.jpg)
The behavior
time
time
time
time
time
SYSTEMSYSTEM
Which event trajectories are possible?
– p.27/40
![Page 63: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/63.jpg)
The behavior
The behavior =all trajectories of the system variableswhich, according to the mathematicalmodel, are possible.
Definition: A system is defined as� �
with � the set of independent variables� the signal spaceQthe behavior.
– p.27/40
![Page 64: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/64.jpg)
The behavior
signal space
time
Totality of ‘legal’ trajectories =: the behavior– p.27/40
![Page 65: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/65.jpg)
Linear shift-invariant differential systems
There exists an extensive theory for these systems
� �
� R the set of independent variables,
� S the set of dependent variables,
� sol’ns lin. const. coeff. system ODE’s or PDE’s.
Let and consider
Define the associated behavior
holds
Notation for n-D linear differential systems:or
– p.28/40
![Page 66: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/66.jpg)
Linear shift-invariant differential systems
There exists an extensive theory for these systems
� �
� R the set of independent variables,often T � �
(dynamical systems)or T � U
(distributed systems),
� S the set of dependent variables,
� sol’ns lin. const. coeff. system ODE’s or PDE’s.
Let and consider
Define the associated behavior
holds
Notation for n-D linear differential systems:or
– p.28/40
![Page 67: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/67.jpg)
Linear shift-invariant differential systems
Let
V W X S Y � . . . R Z and consider
[[]\_^ . . . [[]\!`� � $ � ba �
Define the associated behavior
holds
Notation for n-D linear differential systems:or
– p.28/40
![Page 68: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/68.jpg)
Linear shift-invariant differential systems
Let
V W X S Y � . . . R Z and consider
[[]\_^ . . . [[]\!`� � $ � ba �
Define the associated behavior
� V c R S � � ba �
holds �
Notation for n-D linear differential systems:or
– p.28/40
![Page 69: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/69.jpg)
Linear shift-invariant differential systems
Let
V W X S Y � . . . R Z and consider
[[]\_^ . . . [[]\!`� � $ � ba �
Define the associated behavior
� V c R S � � ba �
holds �
Notation for n-D linear differential systems: R S � V S R or
V S R �
– p.28/40
![Page 70: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/70.jpg)
Examples: Maxwell’s eq’ns, diff. eq’n, wave eq’n, � � �
dfe gih j Bkl mon
dp gih j 9 qq 7 gsrndfe gr j 8n
t u d p gr j Bkl gwvx qq 7 gihzy
(time and space) ,
(electric field, magnetic field,
current density, charge density),
,
the set of solutions to these PDE’s.
Note: 10 variables, 8 equations! free variables.
– p.29/40
![Page 71: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/71.jpg)
Examples: Maxwell’s eq’ns, diff. eq’n, wave eq’n, � � �
de gih j B kl mn
dp gih j 9 qq 7 grndfe gor j 8n
t u dp gsr j B kl gwvx qq 7 g hzy
{ 4 |~} | �
(time and space) �4 �
,� 4 1 ���G ���G ���G � 3 (electric field, magnetic field,
current density, charge density),�4 | � } | � } | � } |G �4 ��
,�� � B 8�4 the set of solutions to these PDE’s.
Note: 10 variables, 8 equations!
�
free variables.– p.29/40
![Page 72: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/72.jpg)
Submodule theorem
V W X W Y � . . . R Z defines� ���� [[\ ^ . . . [[]\�`� �
, but not vice-versa.
¿¿
�
‘intrinsic’ characterization of
V S R � �
Define the annihilators of by
is a -submodule of
Thm. 1: submodules of
– p.30/40
![Page 73: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/73.jpg)
Submodule theorem
V W X W Y � . . . R Z defines� ���� [[\ ^ . . . [[]\�`� �
, but not vice-versa.
¿¿
�
‘intrinsic’ characterization of
V S R � �
Define the annihilators of
V S R by
� �4 ��� � |� ¡ B G H H H G ¡£¢ ¤¥ � ¦ 1 qq¨§© G H H H G qq¨§«ª 3 �4 � ¬®
¯ is a| ¡ B G H H H G ¡°¢ ¤
-submodule of
| � ¡ B G H H H G ¡°¢ ¤
Thm. 1: submodules of
– p.30/40
![Page 74: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/74.jpg)
Submodule theorem
Define the annihilators of
V S R by
� �4 ��� � |� ¡ B G H H H G ¡£¢ ¤¥ � ¦ 1 qq¨§© G H H H G qq¨§«ª 3 �4 � ¬®
¯ is a
| ¡ B G H H H G ¡°¢ ¤-submodule of
| � ¡ B G H H H G ¡°¢ ¤
Thm. 1:S R
�¨± �submodules of
S Y � . . . R Z
– p.30/40
![Page 75: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/75.jpg)
Elimination theorem
Assume
V S^ ² S �R , and define
� - � � � � � such that � � � V
� V S^ R �
Which PDE’s describe ( ) in Maxwell’s equations ?
Eliminate from Maxwell’s equations
– p.31/40
![Page 76: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/76.jpg)
Elimination theorem
Assume
V S^ ² S �R , and define
� - � � � � � such that � � � V
� V S^ R �
Thm. 2 (‘Elimination’ thm): It does!
Which PDE’s describe ( ) in Maxwell’s equations ?
Eliminate from Maxwell’s equations
– p.31/40
![Page 77: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/77.jpg)
Elimination theorem
Which PDE’s describe ( ³ ´ ´
) in Maxwell’s equations ?
Eliminate
´
from Maxwell’s equationsµ H � � 4 �
¶ 8 � G
¶ 8 ··2 µ H � � @ µ H �¸� 4 � G
¶ 8 · E·2 E � �@ ¶ 8 ¹ E µ } µ } ��� @ ··2 �¸� 4 �
– p.31/40
![Page 78: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/78.jpg)
Controllability
[[\ ^ . . . [[]\`� � $
is a ‘kernel representation’ of the associated
V S R.
Another representation: image representation
‘Elimination’ thmWhich systems admit an image representation???
– p.32/40
![Page 79: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/79.jpg)
Controllability
[[\ ^ . . . [[]\`� � $
is a ‘kernel representation’ of the associated
V S R.Another representation: image representation
� [[]\ ^ . . . [[\¸`�º �
‘Elimination’ thm
»½¼ 1 1 qq¨§© G H H H G qq¨§ª 3 3 � � �¢ ¾
Which systems admit an image representation???
– p.32/40
![Page 80: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/80.jpg)
Controllability
[[\ ^ . . . [[]\`� � $
is a ‘kernel representation’ of the associated
V S R.Another representation: image representation
� [[]\ ^ . . . [[\¸`�º �
‘Elimination’ thm
»½¼ 1 1 qq¨§© G H H H G qq¨§ª 3 3 � � �¢ ¾
Which systems admit an image representation???
– p.32/40
![Page 81: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/81.jpg)
Controllability
[[\ ^ . . . [[]\`� � $
is a ‘kernel representation’ of the associated
V S R.Another representation: image representation
� [[]\ ^ . . . [[\¸`�º �
‘Elimination’ thm
»½¼ 1 1 qq¨§© G H H H G qq¨§ª 3 3 � � �¢ ¾
Which systems admit an image representation???
Thm. 3: admits image repr. it is ‘controllable’.– p.32/40
![Page 82: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/82.jpg)
Controllability for ¿À Á
Controllability def’n in pictures:
desired trajectory
undersired trajectory
time
controlled
desired future
undesired pasttransition
time
– p.33/40
![Page 83: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/83.jpg)
Controllability for ¿À Á
Controllability def’n in pictures:
desired trajectory
undersired trajectory
time
controlled
desired future
undesired pasttransition
time
– p.33/40
![Page 84: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/84.jpg)
Controllability for PDE’s
Controllability def’n in pictures:
       Â       Â       Â       Â       Â       Â       Â       Â       Â       Â
à à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à Ãà à à à à à ÃÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä Ä
Å Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å ÅÅ Å Å Å Å Å Å Å
Æ
ÇÇ
È]É
ÊËÊ É
ÈË
� � V
.– p.34/40
![Page 85: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/85.jpg)
Controllability for PDE’s
V
‘patches’ � � V.
Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì ÌÌ Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì Ì
Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í ÍÍ Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í Í
Î Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î ÎÎ Î Î Î Î Î Î
Ï Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÏ Ï Ï Ï Ï Ï ÏÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð ÐÐ Ð Ð Ð Ð Ð Ð Ð
Ñ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÑ Ñ Ñ Ñ Ñ Ñ Ñ ÑÒ
Ó
ÔÔ
Controllability : ‘patch-ability’.
– p.35/40
![Page 86: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/86.jpg)
Controllability for PDE’s
V
‘patches’ � � V.
Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ ÕÕ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ Õ
Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö
× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×× × × × × × ×
Ø Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØØ Ø Ø Ø Ø Ø ØÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù ÙÙ Ù Ù Ù Ù Ù Ù Ù
Ú Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÚ Ú Ú Ú Ú Ú Ú ÚÒ
Ó
ÔÔControllability : ‘patch-ability’.
– p.35/40
![Page 87: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/87.jpg)
Are Maxwell’s equations controllable?
The following equations in the scalar potential and
the vector potential , generate exactly the solutionsto Maxwell’s equations:
proves controllability. Illustrates the connection
controllability potential !
– p.36/40
![Page 88: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/88.jpg)
Are Maxwell’s equations controllable?
The following equations in the scalar potential and
the vector potential
´
, generate exactly the solutionsto Maxwell’s equations:
gih j 9 qq 7 gÜÛ 9 d Ýngr j d p gÞÛngÞv j kl q uq 7 u gÛ 9 kl t u d u g Û x k l t u d ß dfe g Û à x k l qq 7 d Ýn
m j 9 k l qq 7 d e gÞÛ 9 kl d u Ý y
proves controllability.
Illustrates the connection
controllability potential !
– p.36/40
![Page 89: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/89.jpg)
Are Maxwell’s equations controllable?
The following equations in the scalar potential and
the vector potential
´
, generate exactly the solutionsto Maxwell’s equations:
gih j 9 qq 7 gÜÛ 9 d Ýngr j d p gÞÛngÞv j kl q uq 7 u gÛ 9 kl t u d u g Û x k l t u d ß dfe g Û à x k l qq 7 d Ýn
m j 9 k l qq 7 d e gÞÛ 9 kl d u Ý y
proves controllability. Illustrates the connection
controllability
�
potential !– p.36/40
![Page 90: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/90.jpg)
Observability
Observability of the image representation
�4 1 ··½á B G H H H G
··½á ¢
3â
is defined as:
º
can be deduced from ,i.e.,
[[]\ã^ . . . [[]\�`�
should be injective.
Example: Maxwell’s equations do not allow a potentialrepresentation that is observable.
– p.37/40
![Page 91: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/91.jpg)
Observability
Observability of the image representation
�4 1 ··½á B G H H H G
··½á ¢
3â
is defined as:
º
can be deduced from ,i.e.,
[[]\ã^ . . . [[]\�`�
should be injective.
Not all controllable systems admit an observable imagerepr’ion. For T � �
, they do. For T �
, exceptionally.
The latent variable in an image repr’ion may be ‘hidden’.
Example: Maxwell’s equations do not allow a potentialrepresentation that is observable.
– p.37/40
![Page 92: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/92.jpg)
Observability
Observability of the image representation
�4 1 ··½á B G H H H G
··½á ¢
3â
is defined as:
º
can be deduced from ,i.e.,
[[]\ã^ . . . [[]\�`�
should be injective.
Example: Maxwell’s equations do not allow a potentialrepresentation that is observable.
– p.37/40
![Page 93: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/93.jpg)
Sources
Dynamical systems (ODE’s): ä 1985
2-D discrete set of ind. variables: Rocha ä 1990Differential-delay systems:
Glusing-Luerssen, Rocha, Zampieri, VettoriPDE’s: Shankar, Pillai, Oberst, Zerz ä 1995
Generalization from to quaternions:Pereira, Vettori ä 2004
– p.38/40
![Page 94: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/94.jpg)
Thank youThank you
Thank you
Thank you
Thank you
Thank you
Thank you– p.39/40
![Page 95: SYSTEMS THEORY - KU Leuven · SYSTEMS THEORY and PHYSICAL MODELS Jan C. Willems K.U. Leuven, Belgium Universidade de Aveiro, 3 de junho 2004 Œ p.1/40](https://reader034.fdocuments.in/reader034/viewer/2022051811/60273f892cd7000f7d21036b/html5/thumbnails/95.jpg)
Details & copies of the lecture frames are availablefrom/at
http://www.esat.kuleuven.ac.be/ åjwillems
– p.40/40