Study Of Chaos In Induction Machine Drives

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STUDY OF CHAOS IN INDUCTION

MACHINE DRIVE SYSTEM

B.Tech. Project

By

MIRZA ABDUL WARIS BEIGH (10289)

AAKASH AGRAWAL (10288)

GOPAL BHARADWAJ (10265)

MOHAN LAL (09223)

DEPARTMENT OF ELECTRICAL ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY,

HAMIRPUR - 177005 (INDIA)

December, 2013


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STUDY OF CHAOS IN INDUCTION

MACHINE DRIVE SYSTEM

A PROJECT

Submitted in partial ful fi lment of the

requirements for the award for the degree

Of

BACHELOR OF TECHNOLOGY

By




MOHAN LAL (09223)

Under The guidance

Of

Dr. Bharat Bhushan Sharma

DEPARTMENT OF ELECTRICAL ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY,

HAMIRPUR - 177005 (INDIA)

December, 2013


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Copyright NIT HAMIRPUR, 2013


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CANDIDATES DECLARARTION

We hereby certify that the work which is being presented in the project report

entitled STUDY OF CHAOS IN INDUCTION MACHINE DRIVE

SYSTEM, in partial fulfillment of the requirements for the award of degree of

the Bachelor of Technology and submitted in the Department of Electrical

Engineering, National Institute of Technology, Hamirpur H.P. is an authentic

record of our own work carried out during a period from August 2013 to

December 2013 under the supervision of Dr. Bharat Bhushan Sharma,

Assistant Professor, Department of Electrical Engineering, N.I.T. Hamirpur.

The matter presented in this project report has not been submitted by us for the

award of any other degree of this or any other university/institute.

Sd/-




MOHAN LAL (09223)

This is to certify that above statement made by the candidate is correct to the best of

my knowledge.

The project Viva-Voce Examination of the Candidates Mirza Abdul Waris Beigh

(10289), Aakash Agrawal (10288), Gopal Bharadwaj (10265)Mohan Lal (09223) has

been held on____________________.

Date: Sd/-

Dr. Bharat Bhushan Sharma

Assistant Professor, EED

Dr. Bharat Bhushan SharmaProject Supervisor

Electrical Engg. Dept.

----------------------------------

External Examiner


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ACKNOWLEDGEMENT

First things first we find it hard to express my gratefulness to Almighty GOD in words

for bestowing upon us His deepest blessings and providing us with the most wonderful

opportunity in the form of life of a human being and for the warmth and kindness he

has showered upon us.

We feel great pleasure in acknowledging our deepest gratitude to our revered guide

and mentor, Dr. Bharat Bhushan Sharma, Assistant Professor, Electrical

Engineering Department, National Institute of Technology Hamirpur, under whose

firm guidance, motivation and vigilant supervision we succeeded in completing our

work. He infused into us the enthusiasm to work on this topic. His tolerant nature

accepted our shortcomings and he synergized his impeccable knowledge with our

curiosity to learn into this fruitful result.

We would sincerely thank Dr. Ravinder Nath HOD, Electrical Engineering

Department who suggested many related points and is always very helpful and

constructive.

Words are inadequate to express our heartfelt gratitude to our affectionate parents

who have shown so much confidence in us and by whose efforts and blessings we have

reached here.

We would also like to thank all the faculty members of Department of Electrical

Engineering for their continuous moral support and encouragement.

Last but not the least we wish to express heartiest thanks to our friends and colleagues

for their support, love and inspiration.

Date:




MOHAN LAL (09223)


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Abstract

Our work brings attention to the nonlinear dynamics of an induction motor's drive

system with indirect field controlled. To understand the complex dynamics of system,

some basic dynamical properties, such as equilibrium, stability are rigorously derived

and studied. Chaotic attractors are first numerically verified through investigatingphase trajectories, Hopf bifurcation, and Lyapunov Exponents. Furthermore, a new

sliding mode control method is studied to gain the synchronization with different

initial values. It can control the system to an equilibrium point. After the control of

chaos in the system, further variation in parameters is carried out to check for the

events where chaos can creep into the system again. This is verified using the

Lyapunov exponents and the Phase Plots. Numerical simulations are presented to

demonstrate the effectiveness of the proposed controllers.

Keywords: Induction motor, Chaos, Chaos control, Synchronization


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Contents

1 Introduction1

1.1Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...........................11.1.1 Chaos in electric drives11.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.3 Organization of the report . . . . . . . . .. . . . . . . . . . . .. . 2

2. Non Linear Dynamical model of Induction Motor................................ 1

3. Motor Model and Problem Formulation................................................. 2

3.1 Derivation of State Space Form...

3.2 Solution of Equations...

4. Bifurcations & Phase Plot analysis......................................................... 34.1 Phase plots........................................................................................... 3

4.2 Hopf Bifurcations.................................................................................4

5. Lyapunov Exponents................................................................................5

5.1 Introduction to Lyapunov exponent.................................................... 5

5.2 Simulation results................................................................................ 7

6. Sliding Mode Technique......................................................................... 7

6.1 Introduction..........................................................................................7

6.2 Control Scheme...................................................................................8

8. Simulations, Analysis and Result............................................................

8.1 Analysis and Solution of Controller..

8.2 Results after parameter variation

8.2.1 Phase Plot and Bifurcations...

8.2.2 Lyapunov Exponent Plot..

9. Conclusion.................................................................................................. 13

Bio-Data of Candidates..

Appendix...

References....................................................................................................


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List of figures


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List of Abbreviations


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List of symbols


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Introduction

1.1 Overview

The Oxford English Dictionary defines Chaos as Behaviour of a system which is

governed by deterministic laws but is so unpredictable as to appear random, owing toits extreme sensitivity to changes in parameters or its dependence on a large number

of independent variables; a state characterized by such behaviour.

The general perception on chaos is equivalent to disorder or even random. It should be

noted that chaos is not exactly disordered, and its random-like behaviour is governed

by a rule - mathematically, a deterministic model or equation that contains no element

of chance. Actually, the disorder-like or random-like behaviour of chaos is due to its

high sensitivity on initial conditions.

Similar to many terms in science, there is no standard definition of chaos.Nevertheless chaos has some typical features:

Nonlinearity: Chaos cannot occur in a linear system. Nonlinearity is anecessary, but not sufficient condition for the occurrence of chaos. Essentially,

all realistic systems exhibit certain degree of nonlinearity.

Determinism: Chaos must follow one or more deterministic equations that donot contain any random factors. The system states of past, present and future

are controlled by deterministic, rather than probabilistic, underlying rules.

Practically, the boundary between deterministic and probabilistic systems may

not be so clear since a seemingly random process might involve deterministicunderlying rules yet to be found.

Sensitive dependence on initial conditions: A small change in the initial stateof 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 underlying rules.

Aperiodicity: Chaotic orbits are aperiodic, but not all aperiodic orbits arechaotic. Almost-periodic and quasi-periodic orbits are aperiodic, but not

chaotic.

1.1.1 Chaos in Electric Drive Systems

The investigation of chaos in electric drive systems can be categorized as three

themes, namely the analysis of chaotic phenomena, the control of chaotic behaviors,

and the application of chaotic characteristics.

Chaos in electric drive systems was firstly identified in induction drive systems in

1989. That is, the bifurcation of induction motor drives was studied (Kuroe and

Hayashi, 1989), which was actually an extension of the instability analysis of pulse-

width-modulation (PWM) inverter systems. The bifurcation and chaos resulting from

the tolerance-band PWM inverter-fed induction drive system was then investigated

(Nagy, 1994; Suto, Nagy, and Masada, 2000). It was also identified that saddle-nodebifurcation, or even Hopf bifurcation, might occur in induction drive systems under


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indirect field oriented control (Bazanella and Reginatto, 2000) and, consequently, the

control of chaos in induction drive systems was investigated. An attempt was made to

use a neural network stabilizing chaos during speed control of induction drive systems

(Asakura et al., 2000). On the other hand, an attempt was made to use periodic speed

command to stimulate the chaotic motion of induction drive systems (Gao and Chau,

2003a).Without taking power electronic switching into consideration, it was identified that the

permanent magnet (PM) brushless DC drive system could be transformed into a

Lorenz system, which is well known to exhibit a Hopf bifurcation and chaotic

behaviour (Hemati, 1994)The application of chaos in electric drive systems has

focused on the practical use of the control of chaos, including the stabilization of

chaos and the stimulation of chaos. For instance, chaotic vibration in an automotive

wiper system not only decreases the wiping efficiency but also causes harmful

distraction to the drivers (Suzuki and Yasuda, 1998). Thus, the corresponding chaos

was directly stabilized by applying an extended time-delay auto-synchronization

control to its DC drive (Wang and Chau, 2006; Wang and Chau, 2009a). This

approach can be realized experimentally because the armature current of the DCmotor can be easily measured by a Hall sensor and the perturbations on the feed-in

motor voltage can be readily produced by a power converter.

To control the undesirable chaos in the permanent magnet synchronous

motor (PMSM), an adaptive dynamic surface control law was designed by Wei and

his co-partners. However, there are few contributions to a current-driven induction

motor, especially, the dynamical model for a whole induction motor system with

indirect field controlled. While, it is a main drive device in modern industry, and its

nonlinear vibration is catholic. Therefore, it is necessary to study the intrinsic quality

of its nonlinear vibration via nonlinear dynamics theory.

1.2 Objective

Chaos control is inquisitive in how to control the chaotic system to the periodic orbit

or equilibrium point with the original parameters remained or only _ne-tuned, because

the system parameters cannot be changed objectively, or the parameters change

largely must pay a great price. Typical control methods have been proposed to achieve

chaos control. For instance, two methods of chaos control with a small time

continuous perturbation were proposed by Pyragas [16]. Ataei et al. [17] presented achaos synchronization method for a class of uncertain chaotic systems using the

combination of an optimal control theory and an adaptive strategy. Wang and his

coworkers [18] used symbolic dynamics and the automaton reset sequence to identifythe current drive word and obtained the synchronization. Nonlinear and linear

feedback controllers were designed to control and synchronize the chaotic system by

Rafikov et al. [19]. Golovin et al. [20] proposed a global feedback control method

based on measuring the maximum of the pattern amplitude over the domain, which

can stabilize the system. Based on OGY approach, a multiparameter semi-continuous

method was designed to control chaotic behaviour by de Paula and Savi [21]. The

united chaotic systems with uncertain parameters were synchronized based on the

CLF method by Wang et al. [22]. Ataei et al. [23] presented a chaos synchronization

method for a class of uncertain chaotic systems using the combination of an optimal

control theory and an adaptive strategy. Among the control methods, sliding mode

technique (SMT) is one of the best methods. Recently, many contributions have beenpublished (see, for example, [24-28]). To our best knowledge, there is little


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information about control method, which could bridge the chaos control and

synchronization from the literature. And, it is a very valuable theory for its stable and

synchronous operation with the power system.

Considering all the above discussion, there are several advantages which make our

approach attractive, compared with prior works. First, the nonlinear dynamical model

for a whole induction motor system with indirect field controlled is proposed, and thenonlinear dynamics behaviors of the system model are analysed including bifurcation

diagrams, phase plots. Moreover, we present a sliding mode control method. And the

control method is effective to the chaos control and synchronization. Numerical

simulations are demonstrated to the effectiveness of the proposed scheme.

1.3 Organisation of the Report

This work is organised as follows. In Chapter 2 we present the nonlinear dynamical

model of current-driven induction motor expressed in a reference frame rotating at

synchronous speed. Chapter 3 discusses the nonlinear dynamical system and the

problem formulation and its numerical results. Chapter 4 presents the HopfBifurcations and the phase plots of the model presented above. Chapter 5 introduces

the Sliding Mode Technique (SMT) and its method of control. In Chapter 6 a sliding

mode controller is presented. Chapter 7 discusses about the Lyapunov exponents and

the various Lyapunov plots obtained for the system. In Chapter 8 we present the

overall analysis and the results. Finally conclusion and future scope is discussed in

Chapter 10.


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Charter 2

Model of induction Motor


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Chapter-3

Motor Model and Problem formulation

3.1 Derivation of State Space Form

The nonlinear dynamical model of a current-driven induction motor expressed in areference frame rotating at synchronous speed is given as follows:

whereRris rotor resistance,Lris rotor self-inductance,Lmis mutual inductance in a

rotating reference frame, npis the number of pole pairs, slis slipping frequency,Jis

inertia coefficient, TLis load, qris quadrature axis component of the rotor flux, dris

direct axis component of the rotor flux, ris rotor angular speed,Rris rotating

resistance coefficient.

The parameters are introduced as follows:

Therefore, the nonlinear dynamical model of induction motor system with indirect

field controlled can be rewritten as follows:

In speed regulation applications, the indirect field oriented control is usually appliedwith a proportional integral (PI) speed loop, and this control strategy is described as

follows:

Where c^1is the estimate for the inverse rotor time constant c1,is the constant

reference velocity, u02 is the constant reference for the rotor flux magnitude,Kpis the

(1)

(2)

(3)


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proportional of the PI speed regulator,Kiis the integral gains of the PI speed

regulator.

The rotor time constant varies widely in practice IFOC system of IM. One sets

c^1= c1. That is to say, if it has a perfect estimate of the rotor time constant, the

control is tuned; otherwise it is said to be detuned. Therefore, the degree of tuning is

set to k = c^1/c1.Obviously, the controller is tuned and one sets k = 1.

Letx3=ref- r andx4= u3, and thus a new fourth dimensional system can be writtenas follows, based on the model of the whole closed-loop system (2) and the control

strategy (3).

3.2 Solution of the equations.

The equilibria of system (4) can be found by solving the following algebraic

equations:

where, c1= 13:67, c2= 1:56, c3= 0:59, c4= 1176, c5= 2:86, u0

2= 4, kp= 0:001, ki= 1,

k = 1:5, TL= 0:5, ref= 181:1 and the initial state is set to x1= 0,x2 = 0:4,x3= -200,

x4 = 6.

The system has three equilibria, which are respectively described as follows:

O(-0:017; 0:455; 0; 0:304),

E+(-0.0220.182 *i; 0.184 + 0.021*i, 0,0.187 -3:981 * i),

E+(-0.022 + 0.182 *i; 0.184 - 0.021*i, 0,0.187 +3:981 * i),

The system has a unique equilibrium O(-0:017; 0:455; 0; 0:304). Linearize the

system at O, and the Jacobian matrix is obtained as follows:

(4)


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For gaining its eigenvalues, we have

|I-J0|

These eigenvalues at equilibrium O are respectively obtained as follows:

1= 1.65 + 40.39i; 2= 1.65 + 40.39i; 3= -18.98; 4= -13.78

1, 2 are complex conjugate pair and their real parts are positive, and 3and 4 are

negative real numbers. Therefore, the equilibrium O is a saddle point. It is unstable.

The other two equilibrium points E+and E-do not belong to the real space. Thus, it is

not necessary to discuss stability of these points.

The theory of dissipative systems is a basic tool to describe the system characteristics.

And dissipative analysis of system (4) is presented as follows. For system (4), it is

noticed that

Obviously, system (4) can have dissipative structure, with an exponential contractionrate:

That is, a volume element V0is contracted by the flow into a volume element

V0 e- (2c1 + c3 + kpc4c5x2)tin time t. This means that each volume containing the system

orbit shrinks to zero as tat an exponential rate - (2c1+ c3+ kpc4c5x2). Therefore,

all system orbits are ultimately confined to some subset of zero volume, and the

asymptotic motion settles on some attractors.


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Chapter-4

Bifurcations & Phase plot analysis

4.1 Phase Plots

The behaviour of the system can be analysed by observing phase plots, Hopf

bifurcation plots and calculating the value of the maximum Lyapunov exponent. The

detailed description of each of the following is given below.

A phase space is a space in which all possible states of a system are represented, with

each possible state of the system corresponding to one unique point in the phase

space. The concept of phase space was developed in the late 19th century by Ludwig

Boltzmann, Henri Poincare, and Willard Gibbs. A plot of position and momentum

variables as a function of time is sometimes called a phase plot or a phase diagram.

The parameters of the motor are listed as c1= 13.67, c2= 1.56, c3= 0.59, c4= 1176, c5

= 2.86, the parameters of the system are given u02= 4, ref= 181.1 rad/s, TL=0.5, kp=

0.001, ki= 0.5, and k = 1.5.

Phase portraits of chaotic system shown here illustrate the existence of only one-wing.

In the below figures the phase portrait are plotted between various states for the

various parameters listed above. We observe that the states are doesnt have any fixed

equilibrium points and are settled on certain attractors known as strange attractors.

Therefore phase portrait serves as one of the approaches to judge the systems chaotic

Fig(a). Phase plot between X1,X2 and X3.


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Fig(b). Phase plot between X2 and X4.

Fig(c) . Phase plot between X2 and X3.


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Fig(d). Phase plot between X1 and X2

Fig(e). Phase plot between X1,X2 and X4

Fig(f). Phase plot between X2,X3 and X4


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4.2 Hopf Bifurcation

The appearance or the disappearance of a periodic orbit through a local change in the

stability properties of a steady point is known as the Hopf bifurcation. The following

theorem works with steady points with one pair of conjugate nonzero purely

imaginaryeigenvalues.It tells the conditions under which this bifurcationphenomenon occurs.

The term Hopf bifurcation (also sometimes called Poincare-Andronov-Hopf

bifurcation) refers to the local birth or death of a periodic solution (self-excited

oscillation) from an equilibrium as a parameter crosses a critical value.

A bifurcation diagram summarizes the essential dynamics of a system, and thus is a

useful tool to observe its nonlinear dynamical response.

Under the occurrence of Hopf bifurcation, the dynamical system may demonstrate a

complicated behaviorthat is, chaos. To further identify the chaotic behavior, thecalculation of Lyapunov exponents plays an important role

The bifurcation diagram is shown below for different load TL with other parameters

kept constant.

Fig(a). Hopf-Bifurcation plot between T and X1
http://en.wikipedia.org/wiki/Eigenvaluehttp://en.wikipedia.org/wiki/Eigenvalue


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It can clearly be seen that there exist a bifurcation in the values of state variables whenthe range of load is (-4,4).


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Chapter 5

Lyapunov exponents

5.1 Introduction to Lyapunov exponent

If all points in a neighbourhood of a trajectory converge toward the same orbit, the

attractor is a fixed point or a limit cycle. However, if the attractor is strange, any two

trajectoriesx(t) =f t(x0) andx(t) + x(t) =f t(x0+ x0) that start out very close to each

other separate exponentially with time, and in a finite time their separation. attains the

size of the accessible state space.

This sensitivity to initial conditions can be quantified as:

where , the mean rate of separation of trajectories of the system, is called the leading

Lyapunov exponent. In the limit of infinite time the Lyapunov exponent is a global

measure of the rate at which nearby trajectories diverge, averaged over the strange

attractor.

The mean growth rate of the distance

between neighbouring trajectories is

given by the leading Lyapunov exponent which can be estimated for long (but not too

long) time t as

(1)

For notational brevity we shall often suppress the dependence of quantities such as =

(x0, t), x(t)= x(x0, t) on the initial point x0. One can use (1) as is, take a small

initial separation x0, track the distance between two nearby trajectories until ||x(t1)||

gets significantly big, then record t11 = ln(||x(t1)|| / ||x0||), rescale x(t1) by factor

x0/x(t1), and continue ad infinitum, with the leading Lyapunov exponent given by

(2)

Deciding what is a safe linear range, the distance beyond which the separationvector

x(t) should be rescaled, is a dark art.

We can start out with a small x and try to estimate the leading Lyapunov exponent from(2).


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5.2 Simulation Result

To further identify the chaotic behaviour, the calculation of Lyapunov exponents has

been done and the results are plotted below.

The lyapunov exponent have been plotted for load TL= 0.5 and TL= 8.5 , it can clearly

be observed that the highest lyapunov exponent is positive for Tl= 0.5 which indicatesthe systems chaotic behaviour .

Fig(a). Dynamics of lyapunov exponent for T= 0.5

Fig(a). Dynamics of lyapunov exponent for T= 0.5


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Chapter-6

Sliding Mode Technique

6.1 Introduction:

Incontrol theory,sliding mode control, or SMC, is anonlinear control method that

alters thedynamics of anonlinear systemby application of adiscontinuous control

signal that forces the system to "slide" along a cross-section of the system's normal

behaviour. Thestate-feedback control law is not acontinuous function of time.

Instead, it can switch from one continuous structure to another based on the current

position in the state space. Hence, sliding mode control is avariable structure control

method. The multiple control structures are designed so that trajectories always move

toward an adjacent region with a different control structure, and so the ultimate

trajectory will not exist entirely within one control structure. Instead, it willslide

along the boundaries of the control structures. The motion of the system as it slides

along these boundaries is called asliding mode[1]and the geometricallocus consisting

of the boundaries is called thesliding (hyper)surface. In the context of modern control

theory, anyvariable structure system,like a system under SMC, may be viewed as a

special case of ahybrid dynamical system as the system both flows through a

continuous state space but also moves through different discrete control modes.

Figure shows an example trajectory of a system under sliding mode control.The sliding surface is described by , and the sliding mode along the

surface commences after the finite time when system trajectories have reached

the surface. In the theoretical description of sliding modes, the system stays

confined to the sliding surface and need only be viewed as sliding along the

surface. However, real implementations of sliding mode control approximate

this theoretical behaviour with a high-frequency and generally non-

deterministic switching control signal that causes the system to "chatter" in a

tight neighbourhood of the sliding surface. This chattering behaviour is evident

in Figure 1, which chatters along the surface as the system

asymptotically approaches the origin, which is an asymptotically stableequilibrium of the system when confined to the sliding surface.
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Intuitively, sliding mode control uses practically infinitegain to force thetrajectories of adynamic system to slide along the restricted sliding mode

subspace. Trajectories from this reduced-order sliding mode have desirable

properties (e.g., the system naturally slides along it until it comes to rest at a

desiredequilibrium). The main strength of sliding mode control is its

robustness.Because the control can be as simple as a switching between twostates (e.g., "on"/"off" or "forward"/"reverse"), it need not be precise and will

not be sensitive to parameter variations that enter into the control channel.

Additionally, because the control law is not acontinuous function,the sliding

mode can be reached in finite time (i.e., better than asymptotic behaviour).

Under certain common conditions,optimality requires the use ofbangbang

control;hence, sliding mode control describes theoptimal controller for a

broad set of dynamic systems.

6.2 Control Scheme:

In sliding-mode control, the designer knows that the system behaves desirably (e.g., ithas a stableequilibrium)provided that it is constrained to a subspace of its

configuration space.Sliding mode control forces the system trajectories into this

subspace and then holds them there so that they slide along it. This reduced-order

subspace is referred to as a sliding (hyper)surface, and when closed-loop feedback

forces trajectories to slide along it, it is referred to as a sliding mode of the closed-loop

system. Trajectories along this subspace can be likened to trajectories along

eigenvectors (i.e., modes) ofLTI systems;however, the sliding mode is enforced by

creasing the vector field with high-gain feedback. Like a marble rolling along a crack,

trajectories are confined to the sliding mode.

The sliding-mode control scheme involves

1. Selection of a (hyper) surface or a manifold (i.e., the sliding surface) such thatthe system trajectory exhibits desirable behaviour when confined to this

manifold.

2. Finding feedback gains so that the system trajectory intersects and stays on themanifold.

Because sliding mode control laws are notcontinuous,it has the ability to drive

trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is

better than asymptotic).

The sliding-mode designer picks a switching function :RnRm that represents a kind

of "distance" that the states x are away from a sliding surface.

A state x that is outside of this sliding surface has (x)0 A state that is on this sliding surface has (x)=0

The sliding-mode-control law switches from one state to another based on the sign of

this distance. So the sliding-mode control acts like a stiff pressure always pushing in

the direction of the sliding mode where (x)=0. Desirable x(t) trajectories will

approach the sliding surface, and because the control law is notcontinuous (i.e., itswitches from one state to another as trajectories move across this surface), the surface
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is reached in finite time. Once a trajectory reaches the surface, it will slide along it and

may, for example, move toward the x=0 origin. So the switching function is like a

topographic map with a contour of constant height along which trajectories are forced

to move.

The sliding (hyper) surface is of dimension n xm where n is the number of states in xand m is the number of input signals (i.e., control signals) in u. For each control index

1


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Chapter-7

Controller Design

Consider the drive system

Dx =Ax +g(x) (5)

wherex(t)=R4denotes the state vector of the 4-dimensional system,A = R4*4

represents the linear part of the system andg :R4R4is the nonlinear part of the

system. Consideringy(t)=R4as the response of state vector of the 4-dimensional

system, we can rewrite the response system as

Dy=Ay+g(y) (6)

The controller u(t) =R4is added to system (6), so it can be rewritten as:

Dy =Ay +g(y) + u(t) (7)

Here, we define the synchronization errors e = y- x. The aim is to choose a suitable

controller u(t) = R4 such that the states of the master and slave systems can reach

synchronization (i.e., lim||e|| = 0, where||.||is the Euclidean norm).

Now, one sets the controller u(t) as

u(t) = u1(t) + u2(t) (8)

where u1(t)=R4is a compensation controller, and u1(t) =Dx-A(x)-g(x). u2(t)=R4 is a

vector function, and will be designed later. Using (8), response system (7) can berewritten as

De(t) =Ae +g(y)- g(x) + u2(t) (9)

In accordance with the procedure of designing active controller, the nonlinear part of

the error dynamics is eliminated by the following the following input vector:

u2(t)=g(x) - g(y) +Kw(t) (10)

Error system (9) is then rewritten as follows

De(t)=Ae +Kw(t) (11)

whereK = [k1; k2; k3; k4]Tis a constant gain vector and w(t) = R is the control input

that satisfies

(12)

As a choice for the sliding surface, we have


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s(t)=Ce (13)

where C = [c1; c2; c3; c4]Tis a constant vector. For sliding mode method, the sliding

surface and its derivative must satisfy the following conditions.

s(t)=0; s.(t) = 0 (14)

One sets:

s.(t)=CD,e(t)=C(Ae+Kw(t))=0 (15)

To satisfy the above condition, the discontinuous reaching law is chosen as follows

Ds(t) = -psign(s) - rs (16)

Where p > 0, r > 0 are the gains of the controller.

Considering (15) and (16), we have

w(t) = -(CK)-1[C(rI +A)e +psign(s)] (18)

Now, the total control law can be de_ned as follows

u(t) =Dx - Ax - g(y) - K(CK)-1[C(rI +A)e +psign(s)] (19)

Using (19) and (9), the error dynamics can be obtained

De = [A - K(CK)-1C(rI +A)]e - K(CK)-1psign(s) (20)

For the sliding term, a linear system is a bounded input (-K(CK)-1p, whens > 0 and

K(CK)-1p, whens < 0). The system (20) is stable, if |arg(eig([A - K(CK)-1C(rI

+A)]))| > /2. It can be shown that choosing appropriateK, C and r can make the error

dynamics stable. Hence, the synchronization is realized.

Similarly, if the drive system (5) is modified as

Dx=0 (21)

Thus, the response system can be controlled to the initial values of drive system. If the

initial values are changed, the controlling to any stable point can be achieved.


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Chapter-8

Simulations, Analysis and Results

8.1 Analysis and Solution of Controller

The numerical simulation results are carried out to verify the applicability andeffectiveness of the proposed sliding mode control method.

It should be noticed that the controller is in action at t = 10. The ode45 solver of

MATLAB.

Software is applied to solve different equations. By taking the parameters as these in

Section 3, system (4) can be rewritten as:

According to 4.1, we get

Let system (22) with initial conditions [xd1; xd2; xd3; xd4]T= [0; 0:4;-200; 6]Tas a

drive system, and system (22) with initial values [xr1; xr2; xr3; xr4]T= [0:3; 0:5; 0:2;

0:4]Tas a response system. The parameters of the controller are set asK = [-2;-6;-2;-

2]T,C = [5; 5; 5; 5], r = 5, andp = 0:2. This selection of parameters results in

eigenvalues (1;2; 3; 4) = (-2247:3;-14:072;-5;-2:1495) which are located in the

stable region.

According to (19), the control signals are obtained as


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Where e1=xr- xd, e2=yr- yd, e3=zr- zd, e4= wr- wd.

The numerical simulation results are given in Figure 5. One can see, the errors

converge to zero immediately after the controller was applied, which implies that the

chaos synchronization between the two systems is realized. Keep the parameters of

the controller fixed, while set the drive system as system (21) to investigate theeffectiveness of the controller. And we still use system (23) as the controller.

Fortunately, Figure 6 illustrates the response states, which show that the response

states follow initial values of the drive system immediately.

Fig. (a)

Fig. (b)


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Fig. (c)

Fig. (d)

The state variables of the response system in the presence of controller (the controller

u(t) is activated at t = 10)


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8.2 Results after parameter variation.

8.2.1 Phase Plot and Bifurcations

The phase portrait analysis was done again with certain change in parameters which

are given below.

The parameters of the motor are listed as c1= 136.7s-1, c2= 15.6H.s

-1, c3= 0.59s-1, c4

= 1176kg-1_m-2and c5 = 28.6, the parameters of the system are given u0

2= 4A, wref=

181:1rad/s, T =8.5; kp= 0:001, ki= 0:5, and k = 1:5

The phase plots are shown below, these phase plots gives us idea about the behavior

of the system.




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Fig(d). Phase plot between X1,X2 and X3


With the value of load TL > 5 , it was observed that the chaos in the systems

reappears if the value of rotor inductance falls by a certain ratio r.

To analyse the systems behaviour in accordance to the inductance of rotor windings ,

bifurcation diagram is plotted below by varying the values of r.


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Fig(a). Hopf-Bifurcation plot between r and X1




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It is clear from the above figures that the chaos occurs for the values of r within range

(5, 15).

8.2.2 Lyapunov Exponent Plot

The chaotic behavior of the system due to the decrease in the inductance value is

confirmed by plotting the Lyapunov exponents.

Fig(a). Dynamics of lyapunov exponent for r= 10


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Fig(a). Dynamics of lyapunov exponent for r= 20


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APPENDIX

1)Code used for plotting phase portrait.

function[ dx ] = phaseplots( t,x)%This function is used for plotting the phase portraitfor the given%system.

% The function is called using ode45 command andfollowing conditions% were taken into account.% tspan = 0:0.1:100;% options = [];% initvalue = [ 0 .4 -200 6 ];

% The values of the various constant used are givenbelow.

c1 = 13.67*10; c2 = 1.56*10; c3 = 0.59; c4 = 1176; c5 =2.86*10;u= 4; kp = 0.001; k1 = 1; k = 1.5;W=181.1; T= 8.5;

dx=zeros(4,1);

dx(1)= -c1*x(1)+c2*x(4)- (k*c1/u)*x(2)*x(4);dx(2)= -c1*x(2) + c2*u + (k*c1/u)*x(1)*x(4) ;dx(3) = -c3*x(3)-c4*(c5*(x(2)*x(4)-x(1)*u)-T-(c3/c4)*W) ;dx(4) = (k1-kp*c3)*x(3)-kp*c4*(c5*(x(2)*x(4)-x(1)*u)-T-(c3/c4)*W) ;

end

2) Code for plotting hopf bifurcation

function[ dx ] = chaos_bifurcation( t,x,r)%This function is used for plotting the hopf-bifurcationfor the given%system.% The values of the various constant used are givenbelow.% r is that ratio by which the value of the inducatancehas been reduceed.

c1 = 13.67*r; c2 = 1.56*r; c3 = 0.59; c4 = 1176;c5 =

2.86*r ;u= 4; kp = 0.001; k1 = 1; k = 1.5;W=181.1; T= 8.5;


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x1 = 0; x2 = 0.4; x3 = -200; x4 = 6;

dx=zeros(4,1);


end

2.1) Calling the bifurcation function

% Specifying the initail conditions required for runningthe ODE function.tspan=0:0.01:10;initvalue=[0 0.4 -200 6];option=[];figure

% recursively calling the ODE expression for differentvalues of 'r'.forr= 1:.2:30

[t,x]=ode45(@chaos_bifurcation,tspan,initvalue,option,r);hold on;plot(r,x(j,4),'marker','+')xlabel('r')ylabel('X4')

end

3) Code for plotting the lyapunov exponents.

3.1 Main lyapunov function

function[Texp,Lexp]=lyapunov(n,rhs_ext_fcn,fcn_integrator,tstart,stept,tend,ystart,ioutp);%% Lyapunov exponent calcullation for ODE-system.%

% The alogrithm employed in this m-file fordetermining Lyapunov


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% exponents was proposed in%% A. Wolf, J. B. Swift, H. L. Swinney, and J. A.Vastano,% "Determining Lyapunov Exponents from a Time

Series," Physica D,% Vol. 16, pp. 285-317, 1985.%% For integrating ODE system can be used any MATLABODE-suite methods.% This function is a part of MATDS program - toolbox fordynamical system investigation% See: http://www.math.rsu.ru/mexmat/kvm/matds/%% Input parameters:% n - number of equation

% rhs_ext_fcn - handle of function with right handside of extended ODE-system.% This function must include RHS of ODE-system coupled with% variational equation (n items oflinearized systems, see Example).% fcn_integrator - handle of ODE integratorfunction, for example: @ode45% tstart - start values of independent value (timet)% stept - step on t-variable for Gram-Schmidt

renormalization procedure.% tend - finish value of time% ystart - start point of trajectory of ODE system.% ioutp - step of print to MATLAB main window.ioutp==0 - no print,% if ioutp>0 then each ioutp-th point willbe print.%% Output parameters:% Texp - time values% Lexp - Lyapunov exponents to each time value.

%% Users have to write their own ODE functions fortheir specified% systems and use handle of this function asrhs_ext_fcn - parameter.%% Example. Lorenz system:% dx/dt = sigma*(y - x) = f1% dy/dt = r*x - y - x*z = f2% dz/dt = x*y - b*z = f3%

% The Jacobian of system:% | -sigma sigma 0 |


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% J = | r-z -1 -x |% | y x -b |%% Then, the variational equation has a form:%

% F = J*Y% where Y is a square matrix with the same dimensionas J.% Corresponding m-file:% function f=lorenz_ext(t,X)% SIGMA = 10; R = 28; BETA = 8/3;% x=X(1); y=X(2); z=X(3);%% Y= [X(4), X(7), X(10);% X(5), X(8), X(11);% X(6), X(9), X(12)];

% f=zeros(9,1);% f(1)=SIGMA*(y-x); f(2)=-x*z+R*x-y; f(3)=x*y-BETA*z;%% Jac=[-SIGMA,SIGMA,0; R-z,-1,-x; y, x,-BETA];%% f(4:12)=Jac*Y;%% Run Lyapunov exponent calculation:%% [T,Res]=lyapunov(3,@lorenz_ext,@ode45,0,0.5,200,[0 1

0],10);%% See files: lorenz_ext, run_lyap.%% --------------------------------------------------------------------% Copyright (C) 2004, Govorukhin V.N.% This file is intended for use with MATLAB and wasproduced for MATDS-program% http://www.math.rsu.ru/mexmat/kvm/matds/% lyapunov.m is free software. lyapunov.m is distributed

in the hope that it% will be useful, but WITHOUT ANY WARRANTY.% n=number of nonlinear odes% n2=n*(n+1)=total number of odes%

n1=n; n2=n1*(n1+1);

% Number of steps

nit = round((tend-tstart)/stept);

% Memory allocation


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y=zeros(n2,1); cum=zeros(n1,1); y0=y;gsc=cum; znorm=cum;

% Initial values

y(1:n)=ystart(:);

fori=1:n1 y((n1+1)*i)=1.0; end;

t=tstart;

% Main loop

forITERLYAP=1:nit

% Solutuion of extended ODE system

[T,Y] = feval(fcn_integrator,rhs_ext_fcn,[tt+stept],y);

t=t+stept;y=Y(size(Y,1),:);

fori=1:n1forj=1:n1 y0(n1*i+j)=y(n1*j+i); end;

end;

%% construct new orthonormal basis by gram-schmidt%

znorm(1)=0.0;forj=1:n1 znorm(1)=znorm(1)+y0(n1*j+1)^2; end;

znorm(1)=sqrt(znorm(1));

forj=1:n1 y0(n1*j+1)=y0(n1*j+1)/znorm(1); end;

forj=2:n1fork=1:(j-1)

gsc(k)=0.0;forl=1:n1 gsc(k)=gsc(k)+y0(n1*l+j)*y0(n1*l+k);

end;end;

fork=1:n1forl=1:(j-1)

y0(n1*k+j)=y0(n1*k+j)-gsc(l)*y0(n1*k+l);

end;end;


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znorm(j)=0.0;fork=1:n1 znorm(j)=znorm(j)+y0(n1*k+j)^2; end;znorm(j)=sqrt(znorm(j));

fork=1:n1 y0(n1*k+j)=y0(n1*k+j)/znorm(j); end;end;

% update running vector magnitudes

fork=1:n1 cum(k)=cum(k)+log(znorm(k)); end;

% normalize exponent

fork=1:n1lp(k)=cum(k)/(t-tstart);

end;

% Output modification

ifITERLYAP==1Lexp=lp;Texp=t;

elseLexp=[Lexp; lp];Texp=[Texp; t];

end;

if(mod(ITERLYAP,ioutp)==0)fprintf('t=%6.4f',t);fork=1:n1 fprintf(' %10.6f',lp(k)); end;fprintf('\n');

end;

fori=1:n1forj=1:n1

y(n1*j+i)=y0(n1*i+j);end;

end;

end;

3.2 Function where the equations are defined

functionf=lyapunov_chaos(t,X)

% Values of parameters

c1 = 13.67*20; c2 = 1.56*20; c3 = 0.59; c4 = 1176; c5 =2.86*20;


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u= 4; kp = 0.001; k1 = 1; k = 1.5;W = 181.1; T= 8.5;

Y= [X(5), X(9), X(13), X(17);X(6), X(10), X(14), X(18);

X(7), X(11), X(15), X(19);X(8), X(12), X(16), X(20)];

f=zeros(16,1);

% Differential equationf(1)=-c1*X(1)+c2*X(4)-(k*c1/u)*X(2)*X(4);f(2)=-c1*X(2)+c2*u+(k*c1/u)*X(1)*X(4) ;f(3)=-c3*X(3)-c4*(c5*(X(2)*X(4)-X(1)*u)-T-(c3/c4)*W) ;f(4)=(k1-kp*c3)*X(3)-kp*c4*(c5*(X(2)*X(4)-X(1)*u)-T-(c3/c4)*W) ;

%Linearized system

Jac=[ -c1, -(k*c1/u)*X(4), 0, c2-(k*c1/u)*X(2);

(k*c1/u)*X(4), -c1, 0, -(k*c1/u)*X(1);

c4*c5*u, -c4*c5*X(4), -c3, -c4*c5*X(2);

kp*c4*c5*u, -kp*c4*c5*X(4), (k1-kp*c3), -kp*c4*c5*X(2)];

%Variational equationf(5:20)=Jac*Y;

%Output data must be a column vector

3.3 Calling function:

[T,Res]=lyapunov(4,@lyapunov_chaos,@ode45,0,0.5,50,[0 .4-200 6],10);figure

plot(T,Res);title('Dynamics of Lyapunov exponents for T=8.5 and ratiofor C1,C2 % C5 is 20');xlabel('Time');ylabel('Lyapunov exponents');

4) Code used for controller design.

4.1)Uncontrolled system

function[ dx ] = controller( t,x)


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%This function is used for obtaining the state responsefor the given%system.

% The function is called using ode45 command and

following conditions% were taken into account.% tspan = 0:0.1:100;% options = [];% initvalue = [ 0 .4 -200 6 ];

% The values of the various constant used are givenbelow.

c1 = 13.67; c2 = 1.56; c3 = 0.59; c4 = 1176; c5 = 2.86;

u= 4; kp = 0.001; k1 = 1; k = 1.5;W=181.1; T= 0.5;

dx=zeros(4,1);


end

4.2) Controlled system

function[de]= smt_error(t,e)% de(t) = Ae + K*w(t)% calling function. is [t,e]=ode45(@error1,[0:0.5:10],e_initial);

de=zeros(4,1);de(1) = -2256.6339*e(1) + 1.445*e(2) + -0.90157*e(3) +.4667*e(4) - .0067*sign(5 * e(1));de(2) = -6728.8917*e(1) + -9.335*e(2) + -2.7047*e(3) + -3.28*e(4) - .02*sign(5 * e(2));de(3) = 11210.0361*e(1) + 1.445*e(2) + -1.4916*e(3) + -1.093*e(4) - .0067*sign(5 * e(3));de(4) = -2229.511*e(1) + 1.445*e(2) + .0978*e(3) + -1.093*e(4) - .0067*sign(5 * e(4));

end


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4.3) Final calling function used for calling above 2 functions.

initvalue = [ 0 .4 -200 6 ];tspan = 0:0.147:10;

options= [];[t,x]= ode45(@chaos_phaseplots,tspan,initvalue,options);initvalue = [ 0.3 0.5 0.2 0.4 ];[t,y]= ode45(@controller,tspan,initvalue,options);

%finding out the error of the system without controller.e= x-y;

initvalue = e(69,:);[t1,z]= ode45(@smt_error,10:0.1:30,initvalue,options);

% Plotting the results obtained.

plot(t,a(:,4))hold onplot(t1,z(:,4))xlabel('Time T/(sec)')ylabel('e4')title('Synchronization error between 2 systems')


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References

[1] Diyi Chen, Peng Shi and Xiaoyi Ma, Control and synchronization of chaos in an

induction motor system, International Journal of Innovative Computing, Information

and Control, Volume 8, Number 10(B), October 2012.

[2] D. Y. Chen, Y. X. Liu, X. Y. Ma and R. F. Zhang, Control of a class of fractional-

order chaotic systems via sliding mode, Nonlinear Dynamics, vol.67, no.1, pp.893-

901, 2012.

[3] D. Y. Chen, W. L. Zhao, X. Y. Ma et al., No-chattering sliding mode control chaos

in Hindmarsh Rose neurons with uncertain parameters, Computers and Mathematics

with Applications, vol.61, no.8, pp.3161-3171, 2011.

[4] D. Y. Chen, C. Wu, C. F. Liu et al., Synchronization and circuit simulation of a

new double-wing chaos, Nonlinear Dynamics, vol.67, no.2, pp.1481-1504, 2012.

[5] M. Rafikov and J. M. Balthazar, on control and synchronization in chaotic and

hyper chaotic systems via linear feedback control, Communications in Nonlinear

Science and Numerical Simulation, vol.13, no.7, pp.1246-1255, 2008.

[6] A. S. de Paula and M. A. Savi, A multiparameter chaos control method based on

OGY approach, Chaos, Solitons & Fractals, vol.40, no.3, pp.1376-1390, 2009.

[7] H. Wang, Z. Z. Han, W. Zhang and Q. Y. Xie, Synchronization of united chaotic

systems with uncertain parameters based on the CLF, Nonlinear Analysis: Real World

Applications, vol.10, no.2, pp.715-722, 2009.

[8] M. Ataei, A. Iromloozadeh and B. Karimi, Robust synchronization of a class of

uncertain chaotic systems based on quadratic optimal theory and adaptive strategy,

Chaos, vol.20, no.4, pp.043137, 2010. 7248 D. CHEN, P. SHI AND X. MA

[9] B. Jiang, P. Shi and Z. Mao, Sliding mode observer-based fault estimation for

nonlinear networked control systems, Circuits Systems and Signal Processing, vol.30,

no.1, pp.1-16, 2011.

[10] D. Y. Chen, R. F. Zhang, X. Y. Ma et al., Chaotic synchronization and anti-

synchronization for a novel class of multiple chaotic systems via a sliding modecontrol scheme, Nonlinear Dynamics, vol.69, no.1-2, pp.35-55, 2012.

[11] K. T. Chau and Z. Wang. Chaos in electric Drive SystemsAnalysis, Control

and Application. Singapore: Wiley-IEEE Press 2011.

[12] The Wikipedia online encyclopedia:www.wikipedia.com.
http://www.wikipedia.com/http://www.wikipedia.com/http://www.wikipedia.com/http://www.wikipedia.com/

Study Of Chaos In Induction Machine Drives

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