study of chaos in induction machines
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P r e p a r e d B y :
M i r z a A b d u l Wa r i s B e i g h ( 1 0 2 8 9 )
A a k a s h A g g r a w a l ( 1 0 2 8 8 )
G o p a l B h a r a d w a j ( 1 0 2 6 5 )
M o h a n L a l ( 0 9 2 3 3 )
Study of Chaos In Induction Machine Drives
U n d e r t h e g u i d a n c e o f :
D r. B h a r a t B h u s h a n
Major Project Internal Assessment
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.
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
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.
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 long-term 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.
Our approach
Analysis of non linear dynamical
model of the induction machine
Hopf Bifurcations
Phase Plots
Lyapunov Exponents
Model Of Induction Machine
• 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
This model of induction machine was developed by W. Leonhard in 1996.
State Space Model
Here the parameters are as follows:
Hopf-Bifurcations
Between x 1 and TL.
Hopf-Bifurcations(contd…)
Between x 2 and TL.
Hopf-Bifurcations(contd…)
Between x 3 and TL.
Hopf-Bifurcations(contd…)
Between x 4 and TL.
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.
Phase Plots(contd…..)
Variation of x2 w.r.t. x3. Here the system settles to a double wing
type chaotic attractor
Phase Plots(contd…..)
The variation of x2 w.r.t. x4.
Phase Plots(contd…..)
The variation of x1, x2,x4.
Phase Plots(contd…..)
The variation of x2, x3, x4.
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.
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.
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.
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.
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