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

What is Chaos? Deterministic but looks stochastic

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

What is Chaos? Continuous spectrum

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 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)

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 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

Linearization of Poincare Map (OGY) Poincare section

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

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

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.