Chaos ControlChaos ControlPart IIPart II

Chaos ControlChaos ControlPart IIPart II

Amir massoud FarahmandAmir massoud FarahmandSoloGen@SoloGen.netSoloGen@SoloGen.net

Review• Why Chaos control?!

– THE BEGINNING WAS CHAOS!– Chaos is Fascinating!– Chaos is Everywhere!– Chaos is Important!– Chaos is a new paradigm shift in

science!

Review IIWhat is it?!

• Nonlinear dynamics• Deterministic but looks stochastic• Sensitive to initial conditions (positive Bol

(Lyapunov) exponents)

• Strange attractors• Dense set of unstable periodic orbits

(UPO)• Continuous spectrum

Review IIIChaos Control: Goals

• Stabilizing Fixed points• Stabilizing Unstable Periodic Orbits• Synchronizing of two chaotic

dynamics• Anti-control of chaos• Bifurcation control

Review IVChaos Control: Methods

• Linearization of Poincare Map– OGY (Ott-Grebogi-York)

• Time Delayed Feedback Control• Impulsive Control

– OPF (Occasional Proportional Feedback)• Open-loop Control• Conventional control methods

Chaos ControlConventional control

• Back-stepping– A. Harb, A. Zaher, and M. Zohdy, “Nonlinear recursive

chaos control,” ACC2002.

• Frequency domain methods– Circle-like criterion to ensure L2 stability of a

T-periodic solution subject to the family of T-periodic forcing inputs.

– M. Basso, R. Genesio, and L. Giovanardi, A. Tesi, “Frequency domain methods for chaos control,” 2000.

Chaos ControlConventional + Chaotic• Taking advantage of inherit properties of

chaotic systems• Periodic Chaotic systems are dense (according

to Devaney definition)• Waiting for the sufficient time, every point of

the attractor will be visited.• If we are sufficiently close to the goal, turn-on

the conventional controller, else do nothing!– T. Vincent, “Utilizing chaos in control system design,”

2000.

Chaos ControlConventional + Chaotic• Henon map• Stabilizing to the unstable fixed point• Locally optimal LQR design• Farahmand, Jabehdar, “Stabilizing Chaotic Systems with Small

Control Signal”, unpublished

12

2211

3.0)1(

14.1)1(

xkx

uxxkx

otherwise 0

x-x(k) )()(

* kKxku

Figure 1 Henon map

Chaos ControlConventional + Chaotic

0 50 100-0.5

0

0.5

1

k

sta

tes

threshold = 1.0

0 50 100-1.5

-1

-0.5

0

0.5

k

contr

ol eff

ort

threshold = 1.0

0 50 100-2

-1

0

1

2

k

sta

tes

threshold = 0.1

0 50 100-0.02

-0.01

0

0.01

0.02

0.03

k

contr

ol eff

ort

threshold = 0.1

x1

x2

x1

x2

Figure 2. Sample response for two different attraction threshold

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

20

30

40

50

60

70

80

90

100

theta

Sett

ling t

ime

Figure 5. Settling time for different theta

Chaos ControlConventional + Chaotic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

theta

peak o

f u(t

)

Figure 3. Peak of control signal for different theta

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

thetaenerg

y o

f u(t

)

Figure 4. Controlling energy for different theta

Chaos Control• Impulsive control of periodically forced chaotic

system• Z. Guan, G. Chen, T. Ueta, “On impulsive control of

periodically forced pendulum system,” IEEE T-AC, 2000.

Anti-Control of ChaosDefinitions and Applications

(I)• Anti-control of chaos

(Chaotification) is– Making a non-chaotic system,

chaotic.– Enhancing chaotic properties of a

chaotic system.

Anti-Control of Chaos Definition and Applications (II)

• Stability is the main focus of traditional control theory.

• There are some situations that chaotic behavior is desirable– Brain and heart regulation– Liquid mixing– Secure communication– Small control (Chaotification of non-chaotic

system chaos control method (small control) conventional methods )

Anti-Control of Chaos Discrete case (I)

Suppose we have a LTI system. If we change its dynamic with a proper feedback such that it

1. is bounded2. has positive Lyapunov exponent

then we may have made it chaotic.

We may use Marotto theorem to prove the existence of chaos in the sense of Li and Yorke.

X. Wang and G. Chen, “Chaotification via arbitrarily small feedback controls: theory, methods, and applications,” 2000.

Anti-Control of Chaos Discrete case (II)

Anti-Control of Chaos Discrete case (III)

Anti-Control of ChaosContinuous case (I)

• Approximating a continuous system by its time-delayed version (Discrete map).

• Making a discrete dynamics chaotic is easy.

• It has not been proved yet!• X. Wang, G. Chen, X. Yu, “Anticontrol of chaos in

continuous-time systems via time-delayed feedback,” 2000.

Anti-Control of ChaosContinuous case (II)

• Carrier Clock, Secure communication, Power systems and …

• Formulation:

• Synchronization– Unidirectional (Model Reference Control)– Mutual

Synchronization(I)

k1,...,i ,),,...,,(x : 21i txxxFS kii

k1,...,i ,0),,...,( 11txxQ ki k k1,...,i ,0),,...,(lim 11

txxQ kit k

)()(),( 2121 txtxxxQ

Synchronization(II)

• Linear coupling

Synchronization(III)

• Drive-Response concept of Pecora-Carroll

• L.M. Pecora and T.L. Carol, “Synchronization in chaotic systems,” 1990.

Synchronization of Semipassive systems

(I)• A. Pogromsky, “Synchronization and adaptive

synchronization in semipassive systems,” 1997.

• Semipassive Systems

t

t

dxHyuttxVttxV0

))(()(),(),(),( 00

0)(x ;0 xH

Isidori normal form 1,2i ),(

),(),(

iii

iiiiiii

yzqz

uyzbyzay

• Control Signal

211 yyu

,

12 uu

Synchronization of Semipassive systems

(II)• Lemma: Suppose that previous

systems are semipassive with radially unbounded continuous storage function. Then all solutions of the coupled system with following control exist on infinite time interval and are bounded.

0)( yyT

)()( yy

Synchronization of Semipassive systems

(III)• Theorem I: Assume that

– A1. The functions q, a, b are continuous and locally Lipschitz– A2.The system is semipassive– A3.There exist C2-smooth PD function V0 and … that

– A4.The matrix b1+b2 is PD:

– A5.

then there exist … that goal of synchronization is achieved.

2

211211210 ),(),()( zzyzqyzqzzV T

0,),(),( 222111 mIyzbyzb

2)( yyyT

Synchronization of Semipassive systems

(IV)• Lorenz system (Turbulent dynamics of the

thermally induced fluid convection in the atmosphere)

0 5 10 15 20 25-600

-500

-400

-300

-200

-100

0

100

T

erro

r(dB

)

0 5 10 15 20 25-100

-80

-60

-40

-20

0

20

40

60

80

100

T

cont

rol e

ffor

t

Figure 1. error and control signal for linearly coupled system