Chaos theory Study

1. Study of Chaos In Induction
Machine Drives
Major Project Internal Assessment
Under the guidance of:
D r. B h a r a t B h u s h a n
Prepared By:
M i r z a A b d u l Wa r i s B e i g h ( 1 0 2 8 9 )
Aakash Aggrawal (10288)
Gopal Bharadwaj (10265)
Mohan Lal (09233)

2. What is Chaos/Chaos Theory?
 Dictionary meaning: A state of confusion,
lacking any order.
 But in the context of chaos theory, chaos refers to an
apparent lack of order in a system that nevertheless
obeys particular laws or rules.
 Chaos theory is the study of nonlinear dynamics, in
which seemingly random events are actually
predictable from simple deterministic equations.

3. Chaos Theory –Introduction
 Unique properties define a ‘Chaotic System’
 Sensitivity to initial conditions – causing large divergence
in the prediction. But this divergence is not infinite, it
oscillates within bounds.
 Discovered by Ed. Lorentz in Weather Modeling

4. Features of Chaos
 Non-Linearity: Chaos cannot occur in a linear
system. Nonlinearity is a necessary, but not sufficient
condition for the occurrence of chaos. Essentially, all
realistic systems exhibit certain degree of
nonlinearity.
 Determinism: Chaos follows one or more
deterministic equations that do not contain any
random factors. Chaos is not exactly disordered, and
its random-like behaviour is governed by a
deterministic model.

5. Features of Chaos
 Sensitivity to initial conditions: A small change
in the initial state of the system can lead to extremely
different behaviour in its final state. Thus, the longterm prediction of system behaviour is impossible,
even though it is governed by deterministic rules.
 Aperiodicity: Chaotic orbits are aperiodic, but not
all aperiodic orbits are chaotic.

6. Our approach
Hopf
Bifurcations
Analysis of
non linear
dynamical
model of the
induction
machine
Phase Plots
Lyapunov
Exponents

7. Model Of Induction Machine
This model of induction machine was developed by W. Leonhard in 1996.
•
•
•
•
•
•
Rr is rotor resistance
Lr is rotor self-inductance
Lm is mutual inductance
np is the number of pole pairs
ωsl is slipping frequency
J is inertia coefficient
• TL is load
• φqr is quadrature axis component
of the rotor flux.
• φdr is direct axis component of the
rotor flux
• ωr is rotor angular speed
• Rr is rotating resistance coefficient

8. State Space Model
Here the parameters are as follows:

9. Hopf-Bifurcations
 Between x1 and TL.

10. Hopf-Bifurcations(contd…)
 Between x2 and TL.

11. Hopf-Bifurcations(contd…)
 Between x3 and TL.

12. Hopf-Bifurcations(contd…)
 Between x4 and TL.

13. Phase Plots
 This plot shows the variation of x1 w.r.t. x2. As we
can see the
system is
chaotic since
the response
settles into
an attractor.

14. Phase Plots(contd…..)
 Variation of x2 w.r.t. x3. Here the system settles to a
double wing
type chaotic
attractor

15. Phase Plots(contd…..)
 The variation of x2 w.r.t. x4.

16. Phase Plots(contd…..)
 The variation of x1, x2,x4.

17. Phase Plots(contd…..)
 The variation of x2, x3, x4.

18. Lyapunov Exponents
 The Lyapunov exponent can be used to determine
the stability of quasi-periodic(almost periodic) and
chaotic behaviour, and also the stability of
equilibrium points and periodic behaviours.
 The Lyapunov exponent is the exponential rate of the
divergence or convergence of the system states.
 If the maximum Lyapunov exponent of a dynamical
system is positive, this system is chaotic; otherwise,
it is non chaotic.

19. Lyapunov Exponents of this model
In this graph we have
plotted the 4
Lyapunov exponents
of the system. As we
can see one of the
exponents remains
positive and thus the
system is chaotic.
This plot is take by
keeping the value of
load T= 0.5.

20. Removal of chaos from the system
By increasing the
value of the load (T)
upto T=8.5 it was
observed that all the
lyapunov exponents
become sufficiently
negative.
By varying the Load
parameter we were
able to eliminate the
system chaos.

21. Further work
 To Design a controller for the chaotic system using
Sliding mode technique.
 To analyze the variation of parameters so the chaos
of the system can be eliminated.