1. Amir massoud Farahmand SoloGen@SoloGen.net
Chaos Control
Amir massoud Farahmand

2. 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.

3. What is Chaos?
Nonlinear dynamics

4. What is Chaos?
Deterministic but looks stochastic

5. What is Chaos?
Sensitive to initial conditions (positive Bol (Lyapunov) exponents)

6. What is Chaos?
Continuous spectrum

7. 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

8. Chaos Control Chaos is controllable Nonlinear control
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)

9. 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)

10. 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

11. 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.

12. 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

13. 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
To use the discrete system model based on linearization of the Poincare map for controller design.
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

14. Linearization of Poincare Map (OGY)
Poincare section

15. Time-Delayed Feedback Control
Stabilizing T-periodic orbit
K. Pyragas, “Continuous control of chaos be self-controlling feedback,” Phys. Lett. A., vol. 170, 1992.

16. 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)

17. 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

18. 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.

19. 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.

20. 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.