1. Chaos Control (Part III)

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

5. Bifurcation: a case study

6. Bifurcation: a case study

7. Bifurcation: a case study
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) Dynamic
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. Bifurcation Control: Thermal Convection Loop Model
Delaying bifurcation using linear controller

24. Bifurcation Control: Thermal Convection Loop Model

25. 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

28. Bifurcation Control via Normal forms and Invariants

29. Bifurcation Control via Normal forms and Invariants

30. Bifurcation Control via Normal forms and Invariants

31. Bifurcation Control via Normal forms and Invariants

32. A Brief note on Chaotification and Small Control

33. A Brief note on Chaotification and Small Control

34. A Brief note on Chaotification and Small Control
Chaos + Conventional
Control
Conventional Control
Control Energy = 7.06
Max Control = 0.7
Control Energy = 5.40
Max Control = 0.217

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 Synchronization Bifurcation
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.