Transcript CYBERNETICAL PHYSICS - About us | Department of Adaptive

Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
INTRODUCTION TO
CYBERNETICAL PHYSICS
Alexander FRADKOV,
Institute for Problems of Mechanical Engineering
St.Petersburg, RUSSIA
-----------------------------------------------------------------------Prague, UTIA, November 1, 2006
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Laboratory “Control of Complex Systems”
OUTLINE
1. Introduction
2. Features of the control problems in physical systems
3. Results from the “Control of Complex Systems” Lab
3.1. Energy control of conservative systems
3.2. Excitability analysis of dissipative systems
3.3. Examples: Kapitsa pendulum, escape from
potential well;
3.4. Control of molecular systems: classical or quantum?
3.4.1. Dissociation of diatomic molecules
3.4.2. Dissociation of triatomic molecules
3.5.Controlled synchronization of two pendulums
3.6 Excitation of oscillations and waves in a chain of
oscillators.
4. Conclusions
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Laboratory “Control of Complex Systems”
Publications on “Control of chaos” and “Quantum control” in 1990-2004
based on data from Science Citation Index (Web of Science)
800
"Control of Chaos"
"Quantum Control"
700
600
500
400
300
200
100
0
1990
1992
1994
1996
1998
2000
2002
2004
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Laboratory “Control of Complex Systems”
Publications of 1990-2004 in Physical Review A-E, Physical Review Letters
with the term “control” in the title
250
200
150
100
50
0
1990
1992
1994
1996
1998
2000
2002
2004
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Laboratory “Control of Complex Systems”
Publications of 2003:
“Control AND Chaos” - - - - - - - 462
“Control AND Quantum” - - - - 658
Total - 1120
========================================================
IEEE Trans. Autom. Control - - - - - - - - 321
IFAC Automatica - - - - - - - - - - - - - - - - 220
Systems & Control Letters - - - - - - - - - - 107
Intern. Journal of Control - - - - - - - - - - 172
Total - 820
(In Russian – 3 journals, ~350 papers)
****************************************
“Control AND Lasers” - 180
“Control AND Thermodynamics” - - - 79
“Control AND Beams” - 260 “Control AND Plasma AND Tokamaks”-5102
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- There are two fields of application of controlling
friction. Obviously there will be
technological applications for reducing vibration and wear. But controlling friction
experiments can also be used to increase our understanding of the physics of dry friction. For
example, using these methods one can measure the effective friction force as a function of the
sliding. ( Elmer F.J. Phys. Rev. E, V.57, 1998, R490-R4906.)
- We have summarized some recently proposed appications of control methods to problems of
mixing and coherence in chaotic dynamical systems. This is an important problem both for its
own intrinsic interest and also from the point of view of applications. Those methods provide
insights also into the origin of mixing and unmixing behavior in natural systems.
(Sharma A., Gupte, N. Pramana - J. of Physics, V.48, 1997, 231-248. )
- We develop novel diagnostics tools for plasma turbulence based on feedback. This ... allows
qualitative and quantitative inference about the dynamical model of the plasma turbulence.
(Sen A.K., Physics of Plasmas, V.7, 2000, 1759-1766.)
- The aim of the researches is twofold:
-- to create a particular product that is unattainable by conventional chemical means;
-- to achieve a better understanding of atoms and molecules and their interactions.
(Rabitz H. et al., Science, 2000, 288, 824-828.)
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Cybernetical physics - studying physical
systems by cybernetical means
Fields of research:
– Control of oscillations
– Control of synchronization
– Control of chaos, bifurcations,
– Control of phase transitions, stochastic resonance
– Control of mechanical and micromechanical systems
– Optimal control in thermodynamics
– Control of plasma, particle beams
– Control of molecular and quantum systems
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CDC 2001 PLENARY LECTURE:
A new
physics?
John Doyle
Control and Dynamical Systems, Caltech
http://www.cds.caltech.edu/~doyle/
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CDC 2004 PLENARY PANEL DISCUSSION:
Challenges and Opportunities for the Future of Control
Moderator: John Doyle
Panelists: Jean Carlson, Christos Cassandras, P. R. Kumar,
Naomi Leonard, and Hideo Mabuchi
http://control.bu.edu/ieee/cdc04/
Connecting physical processes at multiple
time and space scales in quantum, statistical,
fluid, and solid mechanics, remains not only a
central scientific challenge but also one with
increasing technological implications.
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2. TYPES AND FEATURES OF CONTROL
PROBLEMS IN PHYSICAL SYSTEMS
Plant: x  F ( x, u), x  Rn , u  Rm , y  h( x)  Rl
x – state, u – input (control), y – output (observation).
Type 0: u=const
(parameter optimization,
bifurcation analysis)
Type 1: u=u(t) (program control;
u=Asin(ωt) - vibrational control)
Type 2: u=u(t,y) - feedback control
Features: 1. Control is small: | u(t ) |  ,  is small.
2. Goal is “soft”
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Control goals:
– Excitation
– Synchronization
– Chaotization/
dechaotization
Extension: partial stabilization
Results: transformation laws ( instead of conservation laws)
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3. RESULTS OBTAINED IN CCS Lab:
3.1. Energy control of conservative systems
H
q 
,
p
H
p  
, H  H 0 ( q , p )  H1 ( q , p ) u
q
H 0 ( q , p )  Hamiltonia
n (e ne rgy)of fre esyste m,
H1 ( q , p )  inte ractio
n Hamiltonia
n,
u=u(t) - control (forces, fields, parameters).
H 0 q( t ), p( t )   H* ( t   ).
Control goal :
Problem: Find control algorithm u=U(q,p),
q 
ensuring the control goal for
x ( 0)   , x    .
 p
Difficulties: 1. Control is weak: | u( t ) |  ,   small
2. Nonlocal solutions are needed
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Speed-Gradient (SG) algorithms
System: x  F(x,u,t),
x  R , u  R , t  0 (1)
Goal: Q  x(t),t  0 if t  
(2)
Q(x,t)  0  goal function
n
m
T


SGA : a ) u   Γ  uQ, Γ  Γ  0 (3)
(diff. form)
SGA :
(finite form)

b) u    uQ

where Ψ ( z )T z  0 if z  0
(e.g. Ψ ( z )   z, Ψ ( z )   sign z )
( 4)
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Existing results (Fradkov, 1979, 1985):
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Speed-gradient energy control
1

Goal : Q ( x )   H ( x )  H * 2 , x  [ q , p ] Q  H  H* H
2
Control algorithm:
u   H 0 , H1  H  H* ,
H 0 , H1   0 when x    x : Q( x )  Q0 .
H  x ( t )   H* for all x0   .
Theorem. 1. Let
Then
2. Let
Then either
H0 , H1  0 in a countable set.
H  x ( t )   H * or x ( t )  x , whe re
x    e quilibriu
m of fre esyste m
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Extension: Stabilization of invariants
( h(x)=0 - invariant surface of free system)
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Theorem (Fradkov, Shiriaev et al, 1997)
Let f, g, h and their1st and 2nd derivative s
be continuousand bounded at   x: Q(x) Q0 .
Let l  m, Z(x) h(x)T g(x).
A1. Q (x,u) 0 for u  0 (passivity)
Q (x,u)  0 for Z(x) 0, x  Ω
A2. dimS(x)  l for x  Ω, Q ( x )  0,
where S(x)  span Z(x), Lf Z(x),L2f Z(x), , Lf Z  f T Z
A3. ε  0:  connectedcomponent D is bounded,
where Dε  Ω  x : | det Z(x)T Z(x)| ε
T hen thegoal Q  x (t )   0 is achieved  x (0) Ω .




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3.2. Excitability analysis of dissipative systems
q  H ,
p
H 0
p   H  R( p ), R( p )T
  ( p)  0
q
p
Example. Swinging the damped pendulum
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Upper and lower excitability indices:
____
  (  )  lim sup V  x( t ) ,
(5)
  (  )  lim sup V  x( t ) ,
(6)
t   | u()| 
x ( 0)  0
t   | u()| 
x ( 0)  0
Passivity:
 V(x)  0, ( x )  0 :
t

V ( x ( t ))  V ( x (0))  [ w T u  ( x )]ds
0
V(x) - storage (energy-like) function,
w=w(x) - “passive output”
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Theorem. (Fradkov, 2001)
2
2
Le t α 0 | w |  V ( x )  1 | w |  d ,
2
2
 0 | w |   ( x )  1 | w |
2
 γ
  


   ( )   ( )  m1 
The n  0 
 1 
 ρ0
2

  d

Remark: To prove the left inequality u   signw .
is substituted.
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Special case: Euler-Lagrange systems with dissipation
d
 A(q)q   R(q )   (q)  u,
(1)
dt
R - vector of dissipative forces
1
Total energy: H(q, q)  q T A(q )q   (q )
2
1

Upper and lower excitability indices: E ( ) 

____
  ( )  lim sup H q(t),q(t) ,
(2)
  ( )  __
lim sup H q(t),q(t) ,
(3)
t   |u (  )|
x ( 0 ) 0
t   |u (  )|
x ( 0 ) 0


2
  ( ) ,
Theorem.
If 0     i  A(q)   ,  | q |  R(q )T q    | q |2 , 0   (q)  d . Then
2
2
  
   
      ( )    ( )  m       d
2  
 
Corollary. If R(q)  ρq and ρ  0 , then E  ( )~ C  .
ρ
Remark. Locally optimal control is: u   sign q .

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Excitability of pendular systems:
Simple pendulum:



1 2
2

H  1 ,  1     0 1  cos 
2

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Coupled pendulums
[A.Fradkov, B. Andrievsky, K. Boykov. Mechatronics, V.15 (10), 2005 ]





 
1 2
1 2
k
2
2
H 1 , 1 ,  2 , 2  1   1  cos 1  2   1  cos  2  1   2
2
2
2
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2
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Laboratory set-up
• Mechanical unit;
• Electrical unit (interface init);
• Pentium III personal computer
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3.3. Example 1: Stephenson-Kapitsa pendulum
     sin   u sin  ,
2
0
1 2
2

H     0 1  cos .
2
a) Classical Stephenson-Kapitsa pendulum:
r (t )  A sin  t  u(t )   A 2 sin  t
b) Feedback control: H (t )  H* ,
H*  2mgl.
Speed-gradient algorithm:
u( t )   sign H ( t )  H *  ( t ) sin ( t ) 
  0  H ( t )  H * ,  ( t )     0
  0  lim H ( t ) ~  /   ,  ~  H * .26
2
t 
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Example 2: Control of escape from a potential well
Nonlinear oscillator:
Duffing potential:
     ( )  u
 ( )  
2
2

4
4
Problem: find conditions for escape from a potential well by
means of excitation of minimum intensity| u (t ) | u , u  min
A) Harmonic excitation:
u(t )  u sin t.   0.25  umin  0.21
(H.B. Stewart, J.M.T. Tompson, U. Ueda, A.N. Lansburg,
Physica D, v. 85, 1995, pp. 259-295.)
B) Speed-gradient excitation: u (t )  u sign
Theory:
u   2H* , H*  0.25,   0.25  umin  0.1767
Experiment:
umin  0.122
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Simulation results
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Efficiency of feedback
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3.4. Control of molecular systems - femtotechnologies
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3.4.1. Controlled dissociation of 2-atomic molecules
Classical Morse oscillator:
2
p
 ( q  q0 )

H (q, p) 
 V ( q )   ( q ) E ( t ), V ( q )  D 1  e

2m
Quantum Morse oscillator:
  2

H  
 V (q)   (q) E (t ) ,
 (q)  d q .
2
 2m q

Goal: H q( t ), p( t )   H* , H* dissociation energy
Example: hydrogen fluoride (HF)
D
d  0.7876 , D  6.125eV  0.25 a.u.
a0
2

1
γ  1.1741a0 , a0 
 a.u. of length
2
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me e


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Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
M.Goggin, P.Milonni (LANL). Phys.Rev.A 38 (10), 5174 (1988).
a,c) - classical model; b,d) - quantum model
a,b) E ( t )  E 0 cos( L t )
 L  0.979 0 ,  0  
c, d) E ( t )  E1 cos(1t ) 
 E 2 cos( 2 t ),
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2D
,
m
Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
Control of HF molecules dissociation - classical dynamics
Linear chirping:
2
E ( t )  E 0 sin( 0 t   t )
Speed-gradient:
E ( t )  E0 signp( t )
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Laboratory “Control of Complex Systems”
Control of HF molecules dissociation - quantum dynamics
Linear chirping:
2
E ( t )  E 0 sin( 0 t   t )
Speed-gradient:
E ( t )  E0 signp( t )
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(Ananjevskij M., Fradkov A.,Efimov A., Krivtsov A., PhysCon’03)
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Laboratory “Control of Complex Systems”
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3.4.2. Controlled dissociation of 3-atomic molecule
Aux.problem: Controlled Energy Exchange
– cooling of molecules; - selective dissociation;
– localization of modes; - passage through resonance
H  H 1  H 2  H 12
H  H (q,p,u)
Control goal: H 1 ( t )  H 1 , H 2 ( t )  H 2
(constrained partialstabilization)
*
m in Q1 ( x ,u)
Q2 ( x ,u)  0
 Q )
SG - algorithm: u  - (Q
1
2
*
G eneralization:
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Laboratory “Control of Complex Systems”
Controlled dissociation of 3-atomic molecule
Full Hamiltonian of molecule in external field:
H  H mol  d ( R1 , R2 ) E(t )
Molecular Hamiltonian (Rabitz, 1995; Fujimura, 2000) :
H mol
p12
p22
1



P1 P2  V1 ( R1 )  V2 ( R2 )  V12 ( R1 , R2 )
2m1 2m2 M
c
R1, R2 - displacements of bond length;
P1, P2 - conjugate momenta;
E(t) - controlling field.
p12
 V1 ( R1 )
Control goal: H1 ( t )  H* , where H1 
2m1
Speed-gradient control algorithm: u(t )   signP1 (t )
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3.5. Control of chaos by linearization of Poincare‘ map
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Method of Ott-Grebogi-Yorke (OGY):
The problem is reduced to a standard linear control
problem.
2. Control is switched off when
xk 1  S0
Challenge:
How much time and energy is needed for control?
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3.5. CONTROLLED SYNCHRONIZATION
3.5.1. Model of coupled pendulums
Andrievsky B.R.,Fradkov A.L. Feedback resonance in single and
coupled 1-DOF oscillators // Intern. J of Bifurcation and Chaos, 1999,
N 10, pp.2047-2058.
2

1 (t )  1 (t )   0 sin 1 t   k (1   2 )  u1 t   f1 (t ),

2

2 (t )   2 (t )   0 sin  2 t   k ( 2  1 )  u2 t   f 2 (t )
-  i (t ) (i = 1, 2)  deflection angles;
- ui(t) (i = 1, 2)  controlling torques;
- f 1 , f 2  disturbances;
- k  coupling strength (stiffness of the spring).
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

3.5.2 Design of synchronization algorithm



 
1 2
1 2
k
2
2




H 1 , 1 ,  2 ,  2  1   1  cos 1   2   1  cos  2  1   2
2
2
2
2
Goal function: Q( x)  Q ( x)  (1   )QH ( x)
1
2


where Q ( x )  1   2  ,
2
1
2
QH ( x )  H ( x )  H*  , 0    1  weight
2


u( t )     ( t )  (1   ) H ( t ) 1 ( t )  proportinal form



u( t )    sign   ( t )  (1   ) H ( t ) 1 ( t )  re layform
   1   2 ,  H  H  H * ,   0  gain,0    1
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Total system energy:
1
1
k
2
H 1 , 1 ,  2 ,  2   12   2 1  cos1    22   2 1  cos 2   1   2 
2
2
2
 
1
1
2
2
Q (1 ,  2 )     1   2  ,   {1, 1}
2
2
1
2
Q H ( x )   H ( x )  H *  , x  [ 1 , 1 ,  2 ,  2 ]T .
2
Q( x)  Q ( x)  (1   )QH ( x)
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3.5.3 Synchronization algorithms:


u( t )     ( t )  (1   ) H ( t ) 1 ( t )  proportinal form



u( t )    sign   ( t )  (1   ) H ( t ) 1 ( t )  re layform
   1   2 ,  H  H  H * ,   0  gain ,0    1
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3.5.4 Simulation results
  1,   1  2 - antiphasemotion
 2  10 s 2 ,
k  0.5 s  2 ,
 1s ,
  0.7,
1
H*  20 s  2
49
  1,   1  2 , systemwithloss (   0.1s -1 )
50
  1,   1  2 - inphasemotion
51
  1,   1  2 , systemwithloss (   0.1)
52
3.6. EXCITATION OF OSCILLATIONS AND WAVE
IN THE CHAIN OF OSCILLATORS
3.6.1. Model of chain dynamics
53
Total energy:
Control goal:
SG-control laws:
(1)
(2)
54
3.6.2. Simulation results
Control law (2), ω=1.26, k=2, H*=18.75, N=250
2. γ=0.5, α=0.7 (energy control and synchronization)
55
Space-time Diagram
56
Excitation of oscillations
57
3.6.3. Control of cyclic chain
58
Antiphase oscillations wave
59
Energy and control time histories
60
3.6.4. Control of the chain of oscillators
with incomplete measurements
Nonlinear Luenberger observer
61
Simulation results
62
Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
4. Conclusions
Cybernetical physics - studying physical systems
by cybernetical means
Fields of research:
– Control of oscillations
– Control of synchronization
– Control of chaos, bifurcations
– Control of phase transitions, stochastic resonance
– Optimal control in thermodynamics
– Control of micromechanical, molecular and quantum
systems
63
Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
Publications:
•Fradkov A.L. Exploring nonlinearity by feedback.
Physica D, 128(1999), pp. 159-168.
•Fradkov A.L. Investigation of physical systems by means
of feedback. Automation & Remote Control, 1999, N 3.
•Фрадков А.Л. Кибернетическая физика.
СПб:Наука, 2003.
•Fradkov A.L. Application of cybernetical methods in
physics. Physics-Uspekhi, Vol. 48 (2), 2005, 103-127.
• Fradkov A.L. Cybernetical Physics: From Control of
Chaos to Quantum Control, Springer-Verlag, 2006.
64
Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
65
Institute for Problems of Mechanical Engineering of RAS
Laboratory “Control of Complex Systems”
1st International Conference
PHYSICS and CONTROL (PhysCon 2003)
20–22 Aug. 2003, Saint Petersburg, RUSSIA
2nd International Conference
PHYSICS and CONTROL (PhysCon 2005)
24–26 Aug. 2005, Saint Petersburg, RUSSIA
(200-250 participants, 30-33 countries)
More info at: http://physcon.ru/
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