8/13/2019 Study Of Chaos In Induction Machine Drives
1/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
2/47
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
MIRZA ABDUL WARIS BEIGH (10289)
AAKASH AGRAWAL (10288)
GOPAL BHARADWAJ (10265)
MOHAN LAL (09223)
Under The guidance
Of
Dr. Bharat Bhushan Sharma
DEPARTMENT OF ELECTRICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY,
HAMIRPUR - 177005 (INDIA)
December, 2013
8/13/2019 Study Of Chaos In Induction Machine Drives
3/47
Copyright NIT HAMIRPUR, 2013
8/13/2019 Study Of Chaos In Induction Machine Drives
4/47
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/-
MIRZA ABDUL WARIS BEIGH (10289)
AAKASH AGRAWAL (10288)
GOPAL BHARADWAJ (10265)
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
8/13/2019 Study Of Chaos In Induction Machine Drives
5/47
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:
MIRZA ABDUL WARIS BEIGH (10289)
AAKASH AGRAWAL (10288)
GOPAL BHARADWAJ (10265)
MOHAN LAL (09223)
8/13/2019 Study Of Chaos In Induction Machine Drives
6/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
7/47
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....................................................................................................
8/13/2019 Study Of Chaos In Induction Machine Drives
8/47
List of figures
8/13/2019 Study Of Chaos In Induction Machine Drives
9/47
List of Abbreviations
8/13/2019 Study Of Chaos In Induction Machine Drives
10/47
List of symbols
8/13/2019 Study Of Chaos In Induction Machine Drives
11/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
12/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
13/47
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.
8/13/2019 Study Of Chaos In Induction Machine Drives
14/47
Charter 2
Model of induction Motor
8/13/2019 Study Of Chaos In Induction Machine Drives
15/47
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)
8/13/2019 Study Of Chaos In Induction Machine Drives
16/47
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)
8/13/2019 Study Of Chaos In Induction Machine Drives
17/47
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.
8/13/2019 Study Of Chaos In Induction Machine Drives
18/47
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.
8/13/2019 Study Of Chaos In Induction Machine Drives
19/47
Fig(b). Phase plot between X2 and X4.
Fig(c) . Phase plot between X2 and X3.
8/13/2019 Study Of Chaos In Induction Machine Drives
20/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
21/47
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/Eigenvalue8/13/2019 Study Of Chaos In Induction Machine Drives
22/47
Fig(a). Hopf-Bifurcation plot between T and X2
Fig(a). Hopf-Bifurcation plot between T and X3
Fig(a). Hopf-Bifurcation plot between T and X2
It can clearly be seen that there exist a bifurcation in the values of state variables whenthe range of load is (-4,4).
8/13/2019 Study Of Chaos In Induction Machine Drives
23/47
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).
8/13/2019 Study Of Chaos In Induction Machine Drives
24/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
25/47
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.
http://en.wikipedia.org/wiki/Control_theoryhttp://en.wikipedia.org/wiki/Nonlinear_controlhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Nonlinear_systemhttp://en.wikipedia.org/wiki/Discontinuoushttp://en.wikipedia.org/wiki/State_space_%28controls%29http://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Variable_structure_controlhttp://en.wikipedia.org/wiki/Sliding_mode_control#cite_note-Zinober1990-1http://en.wikipedia.org/wiki/Sliding_mode_control#cite_note-Zinober1990-1http://en.wikipedia.org/wiki/Locus_%28mathematics%29http://en.wikipedia.org/wiki/Variable_structure_systemhttp://en.wikipedia.org/wiki/Hybrid_systemhttp://en.wikipedia.org/wiki/Hybrid_systemhttp://en.wikipedia.org/wiki/Variable_structure_systemhttp://en.wikipedia.org/wiki/Locus_%28mathematics%29http://en.wikipedia.org/wiki/Sliding_mode_control#cite_note-Zinober1990-1http://en.wikipedia.org/wiki/Variable_structure_controlhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/State_space_%28controls%29http://en.wikipedia.org/wiki/Discontinuoushttp://en.wikipedia.org/wiki/Nonlinear_systemhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Nonlinear_controlhttp://en.wikipedia.org/wiki/Control_theory8/13/2019 Study Of Chaos In Induction Machine Drives
26/47
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
http://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Stationary_pointhttp://en.wikipedia.org/wiki/Robust_controlhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Stationary_pointhttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/LTI_systemhttp://en.wikipedia.org/wiki/Hypersurfacehttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Hypersurfacehttp://en.wikipedia.org/wiki/LTI_systemhttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/Stationary_pointhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Bang%E2%80%93bang_controlhttp://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Robust_controlhttp://en.wikipedia.org/wiki/Stationary_pointhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Gain8/13/2019 Study Of Chaos In Induction Machine Drives
27/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
28/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
29/47
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.
8/13/2019 Study Of Chaos In Induction Machine Drives
30/47
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
8/13/2019 Study Of Chaos In Induction Machine Drives
31/47
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)
8/13/2019 Study Of Chaos In Induction Machine Drives
32/47
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)
8/13/2019 Study Of Chaos In Induction Machine Drives
33/47
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.
Fig(d). Phase plot between X1 and X4
Fig(d). Phase plot between X1 and X2
8/13/2019 Study Of Chaos In Induction Machine Drives
34/47
Fig(d). Phase plot between X1,X2 and X3
Fig(d). Phase plot between X2 and X4
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.
8/13/2019 Study Of Chaos In Induction Machine Drives
35/47
Fig(a). Hopf-Bifurcation plot between r and X1
Fig(a). Hopf-Bifurcation plot between r and X2
Fig(a). Hopf-Bifurcation plot between r and X3
8/13/2019 Study Of Chaos In Induction Machine Drives
36/47
Fig(a). Hopf-Bifurcation plot between r and X4
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
8/13/2019 Study Of Chaos In Induction Machine Drives
37/47
Fig(a). Dynamics of lyapunov exponent for r= 20
8/13/2019 Study Of Chaos In Induction Machine Drives
38/47
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;
8/13/2019 Study Of Chaos In Induction Machine Drives
39/47
x1 = 0; x2 = 0.4; x3 = -200; x4 = 6;
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
2.1) 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
8/13/2019 Study Of Chaos In Induction Machine Drives
40/47
% 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 |
8/13/2019 Study Of Chaos In Induction Machine Drives
41/47
% 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
8/13/2019 Study Of Chaos In Induction Machine Drives
42/47
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;
8/13/2019 Study Of Chaos In Induction Machine Drives
43/47
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;
8/13/2019 Study Of Chaos In Induction Machine Drives
44/47
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)
8/13/2019 Study Of Chaos In Induction Machine Drives
45/47
%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);
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
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
8/13/2019 Study Of Chaos In Induction Machine Drives
46/47
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')
8/13/2019 Study Of Chaos In Induction Machine Drives
47/47
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/