CYBERNETICAL PHYSICS

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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
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---------------------- Prague, UTIA, November 1,
2006
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Institute for Problems of Mechanical Engineering
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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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Publications on Control of chaos and Quantum
control in 1990-2004 based on data from Science
Citation Index (Web of Science)
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Institute for Problems of Mechanical Engineering
of RAS Laboratory Control of Complex Systems
Publications of 1990-2004 in Physical Review
A-E, Physical Review Letters with the term
control in the title
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Institute for Problems of Mechanical Engineering
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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- 102
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Institute for Problems of Mechanical Engineering
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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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Institute for Problems of Mechanical Engineering
of RAS Laboratory Control of Complex Systems
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/cdc0
4/
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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Institute for Problems of Mechanical Engineering
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2. TYPES AND FEATURES OF CONTROL PROBLEMS
IN PHYSICAL SYSTEMS
x state, u input (control), y output
(observation).
Type 0 uconst (parameter optimization,

bifurcation analysis) Type 1 uu(t) (program
control uAsin(?t) -
vibrational control) Type 2 uu(t,y) - feedback
control
Features 1. Control is small
is small.
2. Goal is soft
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Institute for Problems of Mechanical Engineering
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Control goals

• Excitation

• Synchronization

• Chaotization/
• dechaotization

Extension partial stabilization Results
transformation laws ( instead of conservation
laws)
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Institute for Problems of Mechanical Engineering
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3. RESULTS OBTAINED IN CCS Lab 3.1. Energy
control of conservative systems
uu(t) - control (forces, fields, parameters).
Control goal
Problem Find control algorithm uU(q,p),
ensuring the control goal for
Difficulties 1. Control is weak
2. Nonlocal solutions are needed
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Speed-Gradient (SG) algorithms
System Goal
goal function
where
(e.g.
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Existing results (Fradkov, 1979, 1985)
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Speed-gradient energy control
Control algorithm
Theorem. 1. Let
Then
2. Let
in a countable set.
Then either
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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)
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3.2. Excitability analysis of dissipative systems
Example. Swinging the damped pendulum
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Upper and lower excitability indices
Passivity
V(x) - storage (energy-like) function, ww(x)
- passive output
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Theorem. (Fradkov, 2001)
Remark To prove the left inequality is
substituted.
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Special case Euler-Lagrange systems with
dissipation
R - vector of dissipative forces
Total energy
Upper and lower excitability indices
__
Theorem.
Corollary.
Remark. Locally optimal control is
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Excitability of pendular systems
Simple pendulum
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Coupled pendulums A.Fradkov, B. Andrievsky, K.
Boykov. Mechatronics, V.15 (10), 2005
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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
a) Classical Stephenson-Kapitsa pendulum
b) Feedback control
Speed-gradient algorithm
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Example 2 Control of escape from a potential well
Nonlinear oscillator
Duffing potential
Problem find conditions for escape from a
potential well by means of excitation of minimum
intensity
A) Harmonic excitation
(H.B. Stewart, J.M.T. Tompson, U. Ueda, A.N.
Lansburg, Physica D, v. 85, 1995, pp. 259-295.)
B) Speed-gradient excitation Theory
Experiment
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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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Institute for Problems of Mechanical Engineering
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3.4.1. Controlled dissociation of 2-atomic
molecules
Classical Morse oscillator
Quantum Morse oscillator
dissociation energy
Example hydrogen fluoride (HF)
a.u. of length
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M.Goggin, P.Milonni (LANL). Phys.Rev.A 38 (10),
5174 (1988).
a,c) - classical model b,d) - quantum model
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Control of HF molecules dissociation - classical
dynamics
Linear chirping
Speed-gradient
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Control of HF molecules dissociation - quantum
dynamics
Linear chirping
Speed-gradient
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(Ananjevskij M., Fradkov A.,Efimov A., Krivtsov
A., PhysCon03)
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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

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Controlled dissociation of 3-atomic molecule
Full Hamiltonian of molecule in external field
Molecular Hamiltonian (Rabitz, 1995 Fujimura,
2000)
R1, R2 - displacements of bond length
P1, P2 - conjugate momenta E(t) -
controlling field.
Control goal
Speed-gradient control algorithm
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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.
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.
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3.5.2 Design of synchronization algorithm
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Total system energy
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3.5.3 Synchronization algorithms
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3.5.4 Simulation results
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3.6. EXCITATION OF OSCILLATIONS AND WAVE IN THE
CHAIN OF OSCILLATORS 3.6.1. Model of chain
dynamics
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Total energy
Control goal
SG-control laws
(1)
(2)
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3.6.2. Simulation results
Control law (2), ?1.26, k2, H18.75, N250 2.
?0.5, a0.7 (energy control and synchronization)
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Space-time Diagram
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Excitation of oscillations
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3.6.3. Control of cyclic chain
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Antiphase oscillations wave
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Energy and control time histories
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3.6.4. Control of the chain of oscillators with
incomplete measurements
Nonlinear Luenberger observer
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Simulation results
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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

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Institute for Problems of Mechanical Engineering
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• 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.
• ??????? ?.?. ??????????????? ??????.
• ????????, 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.

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of RAS Laboratory Control of Complex Systems
1st International Conference PHYSICS and CONTROL
(PhysCon 2003) 2022 Aug. 2003, Saint
Petersburg, RUSSIA 2nd International Conference
PHYSICS and CONTROL (PhysCon 2005) 2426 Aug.
2005, Saint Petersburg, RUSSIA (200-250
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