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

2. Review Why Chaos control?! THE BEGINNING WAS CHAOS!
Chaos is Fascinating!
Chaos is Everywhere!
Chaos is Important!
Chaos is a new paradigm shift in science!

3. Review II What is it?! 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

4. Review III Chaos Control: Goals
Stabilizing Fixed points
Stabilizing Unstable Periodic Orbits
Synchronizing of two chaotic dynamics
Anti-control of chaos
Bifurcation control

5. Review IV Chaos Control: Methods
Linearization of Poincare Map
OGY (Ott-Grebogi-York)
Time Delayed Feedback Control
Impulsive Control
OPF (Occasional Proportional Feedback)
Open-loop Control
Conventional control methods

6. Chaos Control Conventional control
Back-stepping
A. Harb, A. Zaher, and M. Zohdy, “Nonlinear recursive chaos control,” ACC2002.
Frequency domain methods
Circle-like criterion to ensure L2 stability of a T-periodic solution subject to the family of T-periodic forcing inputs.
M. Basso, R. Genesio, and L. Giovanardi, A. Tesi, “Frequency domain methods for chaos control,” 2000.

7. Chaos Control Conventional + Chaotic
Taking advantage of inherit properties of chaotic systems
Periodic Chaotic systems are dense (according to Devaney definition)
Waiting for the sufficient time, every point of the attractor will be visited.
If we are sufficiently close to the goal, turn-on the conventional controller, else do nothing!
T. Vincent, “Utilizing chaos in control system design,” 2000.

8. Chaos Control Conventional + Chaotic
Henon map
Stabilizing to the unstable fixed point
Locally optimal LQR design
Farahmand, Jabehdar, “Stabilizing Chaotic Systems with Small Control Signal”, unpublished
Figure 1 Henon map

9. Chaos Control Conventional + Chaotic
Figure 2. Sample response for two different attraction threshold
Figure 5. Settling time for different theta

10. Chaos Control Conventional + Chaotic
Figure 3. Peak of control signal for different theta
Figure 4. Controlling energy for different theta

11. Chaos Control Impulsive control of periodically forced chaotic system
Z. Guan, G. Chen, T. Ueta, “On impulsive control of periodically forced pendulum system,” IEEE T-AC, 2000.

12. Anti-Control of Chaos Definitions and Applications (I)
Anti-control of chaos (Chaotification) is
Making a non-chaotic system, chaotic.
Enhancing chaotic properties of a chaotic system.

13. Anti-Control of Chaos Definition and Applications (II)
Stability is the main focus of traditional control theory.
There are some situations that chaotic behavior is desirable
Brain and heart regulation
Liquid mixing
Secure communication
Small control (Chaotification of non-chaotic system  chaos control method (small control)  conventional methods )

14. Anti-Control of Chaos Discrete case (I)
Suppose we have a LTI system. If we change its dynamic with a proper feedback such that it
is bounded
has positive Lyapunov exponent
then we may have made it chaotic.
We may use Marotto theorem to prove the existence of chaos in the sense of Li and Yorke.
X. Wang and G. Chen, “Chaotification via arbitrarily small feedback controls: theory, methods, and applications,” 2000.

15. Anti-Control of Chaos Discrete case (II)

16. Anti-Control of Chaos Discrete case (III)

17. Anti-Control of Chaos Continuous case (I)
Approximating a continuous system by its time-delayed version (Discrete map).
Making a discrete dynamics chaotic is easy.
It has not been proved yet!
X. Wang, G. Chen, X. Yu, “Anticontrol of chaos in continuous-time systems via time-delayed feedback,” 2000.

18. Anti-Control of Chaos Continuous case (II)

19. Synchronization (I)
Carrier Clock, Secure communication, Power systems and …
Formulation:
Synchronization
Unidirectional (Model Reference Control)
Mutual

20. Synchronization (II)
Linear coupling

21. Synchronization (III)
Drive-Response concept of Pecora-Carroll
L.M. Pecora and T.L. Carol, “Synchronization in chaotic systems,” 1990.

22. Synchronization of Semipassive systems (I)
A. Pogromsky, “Synchronization and adaptive synchronization in semipassive systems,” 1997.
Semipassive Systems ,
Isidori normal form
Control Signal

23. Synchronization of Semipassive systems (II)
Lemma: Suppose that previous systems are semipassive with radially unbounded continuous storage function. Then all solutions of the coupled system with following control exist on infinite time interval and are bounded.

24. Synchronization of Semipassive systems (III)
Theorem I: Assume that
A1. The functions q, a, b are continuous and locally Lipschitz
A2.The system is semipassive
A3.There exist C2-smooth PD function V0 and … that
A4.The matrix b1+b2 is PD:
A5.
then there exist …  that goal of synchronization is achieved.

25. Synchronization of Semipassive systems (IV)
Lorenz system (Turbulent dynamics of the thermally induced fluid convection in the atmosphere)
Figure 1. error and control signal for linearly coupled system