INTRODUCTION TO CYBERNETICAL PHYSICS

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INTRODUCTION TO INTRODUCTION TO CYBERNETICAL PHYSICS CYBERNETICAL PHYSICS

Alexander FRADKOV, Institute for Problems of Mechanical Engineering

St.Petersburg, RUSSIA

Institute for Problems of Mechanical Engineering of RASLaboratory “Control of Complex Systems”

------------------------------------------------------------------------Prague, UTIA, November 1, 2006

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OUTLINE1. Introduction2. Features of the control problems in physical systems3. 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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100

200

300

400

500

600

700

800

1990 1992 1994 1996 1998 2000 2002 2004

"Control of Chaos""Quantum Control"

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Publications of 1990-2004 in Physical Review A-E, Physical Review Letters with the term “control” in the title

0

50

100

150

200

250

1990 1992 1994 1996 1998 2000 2002 2004

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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 of RASLaboratory «Control of Complex Systems»

- 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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John DoyleControl and Dynamical Systems, Caltech

http://www.cds.caltech.edu/~doyle/

AA new physics?

CDC 2001 PLENARY LECTURE:

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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.

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/

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2. TYPES AND FEATURES OF CONTROL PROBLEMS IN PHYSICAL SYSTEMS


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:

lmn RxhyRuRxuxFx )(,,),,(:Plant

,|)(| tu

x – state, u – input (control), y – output (observation).

2. Goal is “soft”

is small.

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Control goals:– Excitation

– Chaotization/ dechaotization

Extension: partial stabilizationResults: transformation laws ( instead of conservation laws)

– Synchronization


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3. RESULTS OBTAINED IN CCS Lab:3.1. Energy control of conservative systems

upqHpqHHqHp

pHq ),(),(,, 10

n,Hamiltonia ninteractio),(system, free of (energy) nHamiltonia),(

pqHpqH

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u=u(t) - control (forces, fields, parameters). ).()(),( * tHtptqH0Control goal :

Problem: Find control algorithm u=U(q,p), ensuring the control goal for .,)(

pqxx 0

Difficulties: 1. Control is weak: small,|)(| tu 2. Nonlocal solutions are needed

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Speed-Gradient (SG) algorithms

Q(x,t)

tx(t),tQt,Ru,RxF(x,u,t), x mn

0(2)if0

10 )(

)(

)(

4):SGA

30):SGA

form) (finite

form) (diff.

Qub

ΓΓ,QΓua

u

Tu

)sign)(,)(0z if0)(

zzΨzzΨzzΨ T

System:

Goal:goal function

where (e.g.

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Existing results (Fradkov, 1979, 1985):

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Speed-gradient energy control

],[,)()( :Goal * pqxHxHxQ 221

,, *HHHHu 10

HHHQ *

Theorem. 1. Let .)(: when, 010 0 QxQxxHH

Then . allfor )( * 0xHtxH

2. Let 010 HH , in a countable set.

Then either system free of mequilibriu

where,)(or )( *

x

xtxHtxH

Control algorithm:

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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)

.(0) achieved is 0)( goal Then the

|det|: re whebounded, is component connected0 A3.

span re whe,0)(,for dim A2.

,0for 0 )(passivity 0for 0 A1.

.Let .at bounded and continuous be

2 and 1 their andLet

T

T2

T0

ΩxtxQεZ(x)Z(x)xΩD

D : εZ fZ, LZ(x),Z(x),LZ(x), L S(x)

xQΩxlS(x)ΩxZ(x)(x,u)Q

u(x,u)Qg(x)h(x)m, Z(x)l

QQ(x)x:ndsth f, g,

ε

fff

sderivative

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3.2. Excitability analysis of dissipative systems

00

)()(),(, p

pH

pRpRqHp

pHq T

Example. Swinging the damped pendulum

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Upper and lower excitability indices:

(5),)(suplim)(

0)0(|)(|

____txV

xut

(6),)(suplim)(

0)0(|)(|

txV

xut

Passivity: :0)(,0 xV(x)

t

dsxuwxVtxV0

T )]([))0(())((

V(x) - storage (energy-like) function, w=w(x) - “passive output”

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Theorem. (Fradkov, 2001)

21

20

21

20

||)(||

,||)(||Let

wxw

dwxVwα

dm

2

1

2

1 00 ρ

γ)()( Then

Remark: To prove the left inequality is substituted.

.sign wu

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Special case: Euler-Lagrange systems with dissipation )1(,)()()( uqqRqqA

dtd

)()(21 T qqqAq)q(q, H

R - vector of dissipative forcesTotal energy:

Upper and lower excitability indices: ,)(1)(

E

(2)00

,suplim)()(|)(|

____

(t)qq(t),xut

H

(3)00

,suplim)()(|)(|

(t)qq(t),xut

H

__Theorem.

Then00If 2T2 .)(,||)(||,)( dqqqqRqqAi

dm

22

2

)()(

Corollary. .ρ

C)~(E,ρ qρ)q R( then 0 and If

Remark. Locally optimal control is: .qu sign

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Excitability of pendular systems: cos1

21, 22

011 HSimple pendulum:

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Coupled pendulums [A.Fradkov, B. Andrievsky, K. Boykov. Mechatronics, V.15 (10), 2005 ]

222222122112211 2

cos121cos1

21,,,

kH

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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

.cos121

,sinsin

20

2

20

H

u

b) Feedback control:

tAtutAtr sin)(sin)( 2a) Classical Stephenson-Kapitsa pendulum:

.2,)( ** mglHHtH Speed-gradient algorithm:

.~,/~)(lim)(,)(

)(sin)()(sign)(

*

**

HtHtHtH

ttHtHtu

t

2000

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Example 2: Control of escape from a potential wellNonlinear oscillator: u )( Duffing potential:

42)(

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Problem: find conditions for escape from a potential well by means of excitation of minimum intensity min,|)(| uutu A) Harmonic excitation:

21.025.0.sin)( min ututu (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:

sign)( utu 1767.025.0,25.0,2 min** uHHu

122.0min u

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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 moleculesClassical Morse oscillator:

20

21

2

)()(),()()(),( qqeDqVtEqqV

mppqH

Quantum Morse oscillator:

.)(,)()()(2 2

2

qdqtEqqVqm

H

** ,)(),( :Goal HHtptqH dissociation energyExample: hydrogen fluoride (HF)

a.u...,. 2501256787600

eVDaDd

2

2

01

017411em

aa.γe

, a.u. of length

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M.Goggin, P.Milonni (LANL). Phys.Rev.A 38 (10), 5174 (1988).

),cos()cos()( d) c,

,,.

)cos()( b)a,

tEtEtE

mD

tEtE

L

L

2211

00

029790

a,c) - classical model; b,d) - quantum model

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Control of HF molecules dissociation - classical dynamics

)sin()( 200 ttEtE )(sign)( tpEtE 0

Linear chirping: Speed-gradient:

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Control of HF molecules dissociation - quantum dynamics

)sin()( 200 ttEtE )(sign)( tpEtE 0

Linear chirping: Speed-gradient:

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(Ananjevskij M., Fradkov A.,Efimov A., Krivtsov A., PhysCon’03)

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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

)(1221

q,p,uHHHHHH

ion)stabilizat partial ed(constrain :goal Control 2211 )()( ** HtH,HtH

0)( )(min

2

1 :tionGeneralizau,xQ

u,x Q

)(- 21 :algorithm-SG QQu

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Controlled dissociation of 3-atomic moleculeFull 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.

Speed-gradient control algorithm:

Control goal:

)(),( 21 tERRdHH mol

),()()( 122 2112221121

2

22

1

21 RRVRVRVPP

cMmp

mpHmol

)(2

,)( 111

21

1*1 RVmpHwhereHtH

)(sign)( 1 tPtu

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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?

01

whenoff switched is Control 2.Sxk

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3.5. CONTROLLED SYNCHRONIZATION

)()(sin)()(

),()(sin)()(

221222

022

112112

011

tftukttt

tftukttt

3.5.1. Model of coupled pendulums

- )(ti (i = 1, 2) deflection angles;

- ui(t) (i = 1, 2) controlling torques;

- 21 , ff disturbances;

- k coupling strength (stiffness of the spring).

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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222222122112211 2

cos121cos1

21,,,

kH

3.5.2 Design of synchronization algorithm

Goal function: )()1()()( xQxQxQ H

weight102121where

2

221

,)()(

,)(

*HxHxQ

xQ

H

10 gain,0,,

form relay)()()1()()(

form lproportina)()()1()()(

*21

1

1

HH

tttsigntu

ttttu

H

H

H

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222222122112211 2

cos121cos1

21,,,

kH

Total system energy:

.],,,[,)(21)(

}1,1{,21

21),(

T2211

2*

221

221

xHxHxQ

Q

H

)()1()()( xQxQxQ H

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3.5.3 Synchronization algorithms:

10 gain,0,,

form relay)()()1()()(

form lproportina)()()1()()(

*21

1

1

HH

tttsigntu

ttttu

H

H

H

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motion antiphase - 1, 21

2

1

2

22

200.7,

, 1 , 5.0 ,10

sH

s

sk

s

*

3.5.4 Simulation results

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)0.1( loss with system , 1, 121

-s

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motion inphase - 1, 21

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0.1)( loss with system , 1, 21

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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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Control goal:

SG-control laws:(1)(2)

Total energy:

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Control law (2), ω=1.26, k=2, H*=18.75, N=2502. γ=0.5, α=0.7 (energy control and synchronization)

3.6.2. Simulation results

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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 oscillatorswith incomplete measurementsNonlinear Luenberger observer

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Simulation results

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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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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.

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INTRODUCTION TO CYBERNETICAL PHYSICS