Chaos Control

1
Chaos Control

• Amir massoud Farahmand
• SoloGen_at_SoloGen.net

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
• 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.