Chaos Control

Amir massoud FarahmandSoloGen@SoloGen.net

The Beginning was the Chaos Poincare (1892): certain mechanical systems

could display chaotic motion. H. Poincare, Les Methodes Nouvelles de la

Mechanique Celeste, Gauthier-Villars, Paris, 1892. Lorenz (1963):Turbulent dynamics of the

thermally induced fluid convection in the atmosphere (3 states systems)

E. N. Lorenz, “Deterministic non-periodic flow,” J. of Atmos. Sci., vol. 20, 1963.

May (1976): Biological modeling with difference equations (1 state logistic maps)

R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, 1976.

What is Chaos?

Nonlinear dynamics

28,

3

8,10;

rb

xybzz

xyyrxy

xyx

What is Chaos?

Deterministic but looks stochastic

0 500 1000 1500 2000 2500 30000

10

20

30

40

50

60

What is Chaos?

Sensitive to initial conditions (positive Bol (Lyapunov) exponents)

0 500 1000 1500 2000 2500 3000-20

-15

-10

-5

0

5

10

15

20

25Two systems with different initial condition (|e(0)|=1e-5)

What is Chaos? Continuous spectrum

1300 1350 1400 1450 1500 1550 1600 1650 17000

1000

2000

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5000

6000

4950 4960 4970 4980 4990 5000 5010 5020 5030 5040 50500

500

1000

1500

2000

2500

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3500

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4500

5000

What is Chaos?

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

Chaos Control

Chaos is controllable It can become stable fixed point,

stable periodic orbit, … We can synchronize two different

chaotic systems Nonlinear control Taking advantage of chaotic

motion for control (small control)

Different Chaos Control Objectives Suppression of chaotic motion Stabilization of unstable periodic

orbit Synchronization of chaotic systems Bifurcation control

Bifurcation suppression Changing the type of bifurcation (sub-

critical to super-critical and …) Anti-Control of chaos (Chaotification)

Applications of Chaos Control (I)

Mechanical Engineering Swinging up, Overturning vehicles and

ships, Tow a car out of ditch, Chaotic motion of drill

Electrical Engineering Telecommunication: chaotic modulator,

secure communication and … Laser: synchronization and suppression Power systems: synchronization

Applications of Chaos Control (II)

Chemical Engineering Chaotic mixers

Biology and Medicine Oscillatory changes in biological

systems Economics

Chaotic models are better predictors of economical phenomena rather than stochastic one.

Chaos Controlling Methods

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

Time Delayed Feedback Control Impulsive Control

OPF (Occasional Proportional Feedback)

Open-loop Control Lyapunov-based control

Linearization of Poincare Map (OGY)

First feedback chaos control method E. Ott, C. Grebogi, and J. A. York, “Controlling

Chaos,” Phys. Rev. Letts., vol. 64, 1990. Basic idea

1. To use the discrete system model based on linearization of the Poincare map for controller design.

2. To use the recurrent property of chaotic motions and apply control action only at time instants when the motion returns to the neighborhood of the desired state or orbit.

Stabilizing unstable periodic orbit (UPO) Keeping the orbit on the stable manifold

Linearization of Poincare Map (OGY)

}0)(;{ xsxS

S) return tofirst of(point ),( uxPx

ukxPkx ),()1(

Poincare section

)()()1( kBukAxkx

control) ng(stabilizi )()( kCxku

otherwise 0

x-x(k) )()( 0kCxku

-10 -5 0 5 10 15 20 2510

15

20

25

30

35

40

45

Time-Delayed Feedback Control

Stabilizing T-periodic orbit K. Pyragas, “Continuous control of

chaos be self-controlling feedback,” Phys. Lett. A., vol. 170, 1992.

)()()( txtxKtu

Time-Delayed Feedback Control

Recently: stability analysis (Guanrong Chen and …) using Lyapunov method

Linear TDFC does not work for some certain systems T. Ushio, “Limitation of delayed feedback control in

nonlinear discrete-time systems,” IEEE Trans. on Circ. Sys., I, vol. 43, 1996.

Extensions Sliding mode based TDFC

X. Yu, Y. Tian, and G. Chen, “Time delayed feedback control of chaos,” in Controlling Chaos and Bifurcation in Engineering Systems, edited by G. Chen, 1999.

Optimal principle TDFC Y. Tian and X. Yu, “Stabilizing unstable periodic orbits of

chaotic systems via an optimal principle,” Physicia D, 1998. How can we find T (time delay)?

Prediction error optimization method (gradient-based)

Impulsive Control Occasional Feedback Controller

E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: The diode resonator,” Phys. Rev. Lett., vol. 67, 1991.

Stabilizing of the amplitude of a limit cycle Measuring local maximum (minimum) of

the output and calculating its deviation from desired one

Can be seen as a special version of OGY

otherwise 0

y-y(k) y-y(k))(

**Kku

Impulsive Control Partial theoretical work has been done on

justification of OPF Recently methods for impulsive control and

synchronization of nonlinear systems have been developed based on theory of Impulsive Differential Equations

V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Pub. Co., 1990.

T. Yang and L. O. Chua, “Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication,” Int. J. of Bifur. Chaos, vol. 7, 1997.

Open-loop Control of Chaotic Systems

Change the behavior of a nonlinear system by applying an external excitation.

Suppressing or exciting chaos Simple Ultra fast processes States of the system are not measurable

(molecular level) General feedforward control method for

suppression or excitation of chaos has not devised yet.

Lyapunov-based methods Most of mentioned methods have

some Lyapunov-based argument of their stability.

More classical methods Speed Gradient Method

A.L. Fradkov and A.Y. Pogromsky, “Speed gradient control of chaotic continuous-time systems,” IEEE Trans. Circuits Syst. I, vol. 43,1996.