1. Chaos Control (Part III) Amir massoud Farahmand SoloGen@SoloGen.net Advisor: Caro Lucas

2. Bifurcation: introduction What is Bifurcation?! Structural change in dynamical system’s properties –Equilibrium set topology –Stability of equilibrium set –Type of dynamic behaviors Equilibrium sets Limit cycles Chaotic

3. Bifurcation: a case study Logistic map –A nonlinear population model Different dynamical behaviors

4. Bifurcation: a case study

7. Bifurcation diagram for Logistic map

8. Bifurcation: applications Bifurcation occurs in –Power systems –Aircraft stall –Aero engines –Road vehicle under steering control –Dynamics of Ships –Cellular Neural Networks –Automatic Gain Control circuits –Double pendulum –…

9. Bifurcation Bifurcation is a route to chaos (during Period-doubling bifurcation or …) Bifurcation can be a result of –Change of parameter –Control signal

10. Bifurcation types Stationary (static) –Topological change of equilibrium sets –Transcritical bifurcation –Saddle-node bifurcation –Pitchfork bifurcation Dynamic –Different dynamical behavior –Hopf bifurcation

11. Bifurcation types: stationary

12. Bifurcation types: dynamic (Hopf)

13. Bifurcation: Hopf theorem

14. Bifurcation Control Different control objectives: Postponing bifurcation Change bifurcation –Stability –Type (from Sub-critical to Super-critical) –Amplitude –Frequency of limit cycle G. Chen, J. Moiola, and H. Wang, “Bifurcation control: Theories, Methods, And Applications,” IJBC, 2000.

15. Bifurcation Control Stability analysis of a nonlinear system –Local linearization of system LHP (stable) RHP (unstable) On imaginary axis: what can be said? –Critical cases Bifurcation Control –What is the system behavior when it confront a bifurcation? E. Abed and J. Fu, “Local feedback stabilization and bifurcation control, I. Hopf Bifurcation,” System and Control Letters, 1986. E. Abed and J. Fu, “Local feedback stabilization and bifurcation control, II. Stationary Bifurcation,” System and Control Letters, 1987.

16. Bifurcation Control Local Stabilization problem (and so, bifurcation behavior) of a nonlinear dynamics can be solved using Center Manifold Theorem. –Obtaining Invariant Manifold is not an easy task. We can remove critical cases by linear feedback –What about the uncontrollable critical case? This is the real interesting problem!

17. Bifurcation Control: Hopf Assumption (H): A(0) has a pair of simple complex conjugate eigenvalues on imaginary axis There exist a limit cycle with following characteristic exponent

18. Bifurcation Control: Hopf If β<0, limit cycle is stable. So finding a method to calculate β will solve bifurcation analysis.

19. Bifurcation Control: Hopf We can change the stability of the system by finding Appropriate values of quadratic And cubic terms of control signal

20. Bifurcation Control: Hopf We can change bifurcation stability by quadratic and cubic terms, even if the system is critically uncontrollable. Linear term can be used to change the place of bifurcation (if it is controllable).

21. Bifurcation Control: Thermal Convection Loop Model H. Wang and E. Abed, “Bifurcation Control of Chaotic Dynamical System,” 1993.

22. Bifurcation Control: Thermal Convection Loop Model

23. Delaying bifurcation using linear controller

24. Bifurcation Control: Thermal Convection Loop Model

26. Bifurcation Control via Normal forms and Invariants W. Kang, “Normal forms, invariants, and bifurcations of Nonlinear control systems,”

27. Bifurcation Control via Normal forms and Invariants

32. A Brief note on Chaotification and Small Control

34. Conventional Control Control Energy = 7.06 Max Control = 0.7 Control Energy = 5.40 Max Control = 0.217 Chaos + Conventional Control

35. What has been told? Properties of Chaos –Nonlinear, Deterministic but looks stochastic, Sensitive, Continuous spectrum, Strange attractors Different possible control objective in chaos control –Suppression, Stabilization of UPO, Synchronization, Chaotification, Bifurcation Control Applications of Chaos

36. What has been told? Chaos Control –OGY –Time-delayed Feedback Control (TDFC) –Impulsive Control (OPF) –Open loop control –Conventional methods (frequency domain, back-stepping, Conventional + Chaos properties –…

37. What has been told? Chaotification –Discrete –Continuous Synchronization –Drive-Response idea –Passivity based Bifurcation –Definition –Some theories and …

38. Last words I have done a survey on chaos control and related fields. Despite my early though of having rather small chaos control literature, it has a large number of published papers. So this survey is not a complete one at all. Anyway, I tried to take a brief look at everything related to chaos.

39. One more last word! Doing a survey on chaos control is very difficult job, because chaos researchers are not confined to a one or two branches of science. Researchers of chaos control might be from pure and applied math., physics, control theory, communication engineer, power system engineers, and …. Beside that, they use a lot of different nonlinear analysis tools that some of them is not familiar to a control theory graduate student.

40. That’s all folks!