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.