Chaos Control(Part III)

Amir massoud Farahmand

SoloGen@SoloGen.net

Advisor: Caro Lucas

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

Bifurcation: a case study

• Logistic map– A nonlinear population model

• Different dynamical behaviors

Bifurcation: a case study

0 2 4 6 8 10 12 14 16 18 200

0.1

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K

X(k

)

Logistic map for p=0.5

0 20 40 60 80 100 120 140 160 180 2000.305

0.31

0.315

0.32

0.325

0.33

0.335

K

X(k

)

Logistic map for p=1.5

Bifurcation: a case study

0 10 20 30 40 50 60

0.6

0.65

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0.75

K

X(k

)

Logistic map for p=3.1

0 5 10 15 20 25 30 35 40

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K

X(k

)

Logistic map for p=3.5

Bifurcation: a case study

0 20 40 60 80 100 120 140 160 180 2000.3

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1

K

X(k

)

Logistic map for p=3.56

Bifurcation: a case study

Bifurcation diagram for Logistic map

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– …

Bifurcation

• Bifurcation is a route to chaos (during Period-doubling bifurcation or …)

• Bifurcation can be a result of – Change of parameter– Control signal

Bifurcation types

• Stationary (static)– Topological change of equilibrium sets– Transcritical bifurcation– Saddle-node bifurcation– Pitchfork bifurcation

• Dynamic– Different dynamical behavior– Hopf bifurcation

Bifurcation types: stationary

Bifurcation types: dynamic (Hopf)

Bifurcation: Hopf theorem

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.

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.

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!

Bifurcation Control: Hopf

)(xFx

...)( 44

22

x

xFA

)(

)( 0

Assumption (H): A(0) has a pair of simple complex conjugate eigenvalues on imaginary axis

There exist a limit cycle with following characteristic exponent

Bifurcation Control: Hopf

...)( 44

22

If β<0, limit cycle is stable.So finding a method to calculate β will solve bifurcationanalysis.

Bifurcation Control: Hopf

We can change the stabilityof the system by finding Appropriate values of quadraticAnd cubic terms of control signal

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).

Bifurcation Control: Thermal Convection Loop Model

H. Wang and E. Abed, “Bifurcation Control of Chaotic Dynamical System,” 1993.

Bifurcation Control: Thermal Convection Loop Model

Bifurcation Control: Thermal Convection Loop Model

• Delaying bifurcation using linear controller

Bifurcation Control: Thermal Convection Loop Model

Bifurcation Control: Thermal Convection Loop Model

Bifurcation Control via Normal forms and Invariants

W. Kang, “Normal forms, invariants, and bifurcations ofNonlinear control systems,”

Bifurcation Control via Normal forms and Invariants

Bifurcation Control via Normal forms and Invariants

Bifurcation Control via Normal forms and Invariants

Bifurcation Control via Normal forms and Invariants

Bifurcation Control via Normal forms and Invariants

A Brief note on Chaotification and Small Control

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

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t

X1

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State variables

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

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1

t

u

Control effort response

0 0.005 0.01 0.015 0.02 0.0250

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X1

X2

State trajectory

A Brief note on Chaotification and Small Control

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t

X1

and

X2

State variables

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.4

-0.2

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t

u

Control effort response

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-0.2

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X1

X2

State trajectory

A Brief note on Chaotification and Small Control

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.1

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X1

and

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State variables

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.8

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t

u

Control effort response

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06-0.1

0

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X1

X2

State trajectory

Conventional Control

Control Energy = 7.06Max Control = 0.7

Control Energy = 5.40Max Control = 0.217

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.1

0

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X1

and

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State variables

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.5

0

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t

u

Control effort response

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X2

State trajectory

Chaos + Conventional Control

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

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

– …

What has been told?

• Chaotification– Discrete– Continuous

• Synchronization– Drive-Response idea– Passivity based

• Bifurcation– Definition– Some theories and …

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.

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.

That’s all folks!