chaos control amir massoud farahmand sologen@sologen.net
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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
3000
4000
5000
6000
4950 4960 4970 4980 4990 5000 5010 5020 5030 5040 50500
500
1000
1500
2000
2500
3000
3500
4000
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.
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