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Journal of the Franklin Institute 348 (2011) 300314

A robust vector control for induction motor drives

with an adaptive sliding-mode control law

Oscar Barambonesa,, Patxi Alkortab

aDpto. Ingeniera de Sistemas y Autom atica, EUI de Vitoria, Universidad del Pas Vasco, Nieves cano 12, 01006

Vitoria, SpainbDpto. Ingeniera de Sistemas y Autom atica, EUI de Eibar, Universidad del Pas Vasco, Avda. Otaola, 29 20600

Eibar (Gipuzkoa)

Received 4 January 2010; received in revised form 24 November 2010; accepted 30 November 2010

Available online 7 December 2010

Abstract

A novel adaptive sliding-mode control system is proposed in order to control the speed of an

induction motor drive. This design employs the so-called vector (or field oriented) control theory forthe induction motor drives. The sliding-mode control is insensitive to uncertainties and presents an

adaptive switching gain to relax the requirement for the bound of these uncertainties. The switching

gain is adapted using a simple algorithm which does not imply a high computational load. Stability

analysis based on Lyapunov theory is also performed in order to guarantee the closed loop stability.

Finally, simulation results show not only that the proposed controller provides high-performance

dynamic characteristics, but also that this scheme is robust with respect to plant parameter variations

and external load disturbances.

& 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The induction motor is a complex structure that converts electrical energy into mechanical

energy. Although induction machines were introduced more than a hundred years ago, the

research and development in this area appears to be never-ending. Traditionally, AC machines

with a constant frequency sinusoidal power supply have been used in constant-speed

applications, whereas DC machines were preferred for variable speed drives, since they present

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0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2010.11.008

Corresponding author. Tel.: 34 945013235; fax: 34 945013270.E-mail address: [email protected] (O. Barambones).
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a simpler control. Besides, AC machines presented some disadvantages in comparison with

DC ones, as higher cost, higher rotor inertia and maintenance problems. Nevertheless, in the

last two or three decades we have seen extensive research and development efforts in variable-

frequency, variable-speed AC machine drives technology[1], which have overcome some of the

above disadvantages of the AC motors.The development of field oriented control in the beginning of 1970s made it feasible to

control the induction motor as a separately excited DC motor[13]. In this sense, the field-

oriented technique guarantees the decoupling of torque and flux control commands for the

induction motor. This means that when the flux is governed by means of controlling the

currentid, the torque is not affected. Similarly, when the torque is governed by controlling

the current iq, the flux is not affected and, therefore, it can be achieved transient response

as fast as in the case of DC machines.

On the other hand, when dealing with indirect field-oriented control of induction

motors, a knowledge of rotor speed is required in order to orient the injected stator current

vector and to establish an adequate speed feedback control. Although the use of a flux

estimator in direct field oriented control eliminates the need of the speed sensor in order to

orient the injected stator current vector, this method is not practical. This is because the

flux estimator does not work properly in a low speed region. The flux estimator presents a

pole on the origin of the Splane (pure integrator), and therefore it is very sensitive to the

offset of the voltage sensor and the parameter variations.

However, the speed or position sensor of induction motor still limits its applications to

some special environments not only due to the difficulties of mounting the sensor, but also

because of the need of low cost and reliable systems. The research and development work on

a sensorless driver for the AC motor is progressing greatly. Much work has been done usingthe field oriented based method approach[47]. In these schemes the speed is obtained based

on the measurement of stator voltages and currents. On the other hand, the induction motor

model can be obtained using a Neural Network approach. In the work of Alanis et al. [8]

a discrete-time nonlinear system identification via recurrent high order neural networks is

proposed. In this work a sixth-order discrete-time induction motor model in the stator fixed

reference frame is calculated using the proposed recurrent neural networks scheme.

Nevertheless, the robustness to parameter variations and load disturbances in the

induction machines still deserves to be further studied and, in particular, special attention

should be paid to the low speed region transients.

Thus, the performance of the field oriented control strongly depends on uncertainties,which are usually due to unknown parameters, parameter variations, external load

disturbances, unmodelled and nonlinear dynamics, etc. Therefore, many studies have been

made on the motor drives in order to preserve the performance under these parameter

variations and external load disturbances, such as nonlinear control, optimal control,

variable structure system control, adaptive control, neural control and fuzzy control

[913]. Recently, the genetic algorithm approach has also been used in order to control the

electric motors. The work of Montazeri-Gh et al. [14], describes the application of the

genetic algorithm for the optimization of the control parameters in parallel hybrid electric

vehicles driven by an electric induction machine.

To overcome the above system uncertainties, the variable structure control strategy usingthe sliding-mode has been focussed on many studies and research for the control of the AC

servo drive system in the past decade[1519]. The sliding-mode control can offer many good

properties, such as good performance against unmodelled dynamics, insensitivity to parameter

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variations, external disturbance rejection and fast dynamic response[20]. These advantages of

the sliding-mode control may be employed in the position and speed control of an AC servo

system.

The robust properties of the sliding-mode systems are also been employed in the

observers design [21]. In this work an observer-based sliding-mode control problem isinvestigated for a class of uncertain delta operator systems with nonlinear exogenous

disturbance and the control system stability is demonstrated using the Lyapunov stability

theory. In the work of Boiko[22]the estimation precision and bandwidth of sliding-mode

observers are analyzed in the frequency domain for different settings of the observer design

parameters. In this paper an example of sliding-mode observer design for estimation of DC

motor speed from the measurements of armature current is considered.

A position-and-velocity sensorless control for brushless DC motors using an adaptive

sliding mode observer is proposed in Furuhashi[23]. In this work a sliding-mode observer

is proposed in order to estimate the position and velocity for brushless DC motors. Then,

the velocity of the system is regulated using a PI control. A sensorless sliding-mode torque

control for induction motors used in hybrid electric vehicle applications is developed in

Proca et al.[24]. The sliding-mode control proposed in this work allows for fast and precise

torque tracking over a wide range of speed. The paper also presents the identification and

parameter estimation of an induction motor model with varying parameters. In the paper

[25]a survey of applications of second-order sliding-mode control to mechanical systems is

presented. In this paper different second-order sliding-mode controllers, previously

presented in the literature, are shown and some challenging control problems involving

mechanical systems are addressed and solved. A robust sliding-mode sensorless speed-

control scheme of a voltage-fed induction motor is proposed in Rashed et al. [26]. In thiswork a second-order sliding mode is proposed in order to reduce the chattering problem

that usually appears in the traditional sliding-mode controllers. In the work of Aurora and

Ferrara [27] a sliding-mode control algorithm for current-fed induction motors is

presented. In this paper is proposed an adaptive second-order sliding-mode observer for

speed and rotor flux, and the load torque and the rotor time constant are also estimated.

The higher order sliding mode (HOSM) proposed in this work, present some advantages over

standard sliding-mode control schemes, one of the most important is the chattering reduction.

However in the HOSM an accurate knowledge of rotor flux and machine parameters is the key

factor in order to obtain a high-performance and high-efficiency induction-motor control

scheme. Then, these control schemes require a more precise knowledge of the system parametersor the use of estimators in order to calculate the system parameters, which implies more

computational cost than traditional sliding-mode controllers.

On the other hand, the sliding control schemes require prior knowledge of the upper bound

for the system uncertainties since this bound is employed in the switching gain calculation.

It should be noted that the choice of such bound may not be easily obtained due to the

complicated structure of the uncertainties in practical control systems [28,29]. Moreover, this

upper bound should be determined as accurately as possible, because the value to be

considered for the sliding gain increases with the bound, and therefore the control effort will be

also proportional to this bound. Hence, a high upper bound for the system uncertainties

implies more control effort and the problem of the chattering will be increased.In order to surmount this drawback, in this paper is proposed an adaptive law in order

to calculate the sliding gain. Therefore, in our proposed adaptive sliding-mode control

scheme we do not need to calculate an upper bound of the system uncertainties, which

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greatly simplifies the controller design. Moreover, this upper bound can be unknown and

can be variable along the time because the sliding gain is adapted on-line.

In this sense, this paper presents a new sensorless vector control scheme consisting on

the one hand of a speed estimation algorithm so that there is no need for a speed sensor

and on the other hand of an adaptative variable structure control law with an integralsliding surface that compensates for the uncertainties in the system. In the proposed

adaptive sliding-mode control scheme, unlike the traditional sliding-mode control schemes,

the sliding gain is not calculated in advance, because it is estimated on-line in order to

compensate the present system uncertainties that can be variables along the time.

Using this variable structure control in the induction motor drive, the controlled speed is

insensitive to variations in the motor parameters and load disturbances. This variable

structure control provides a good transient response and exponential convergence of the

speed trajectory tracking despite parameter uncertainties and load torque disturbances.

The closed loop stability of the proposed scheme is demonstrated using Lyapunov

stability theory, and the exponential convergence of the controlled speed is also provided.

This report is organized as follows. The rotor speed estimation is introduced in Section 2.

Then, the proposed robust speed control with adaptative sliding gain is presented in Section 3.

In Section 4, some simulation results are presented. Finally, concluding remarks are stated in

the last section.

2. Rotor speed computation

Many schemes based on simplified motor models have been devised to sense the speed of

the induction motor from measured terminal quantities for control purposes. In order toobtain an accurate dynamic representation of the motor speed, it is necessary to base the

calculation on the coupled circuit equations of the motor.

Since the motor voltages and currents are measured in a stationary frame of reference, it

is also convenient to express these equations in that stationary frame.

From the stator voltage equations in the stationary frame it is obtained [3]:

_cdr Lr

Lmvds

Lr

LmRs sLs

d

dt

ids 1

_cqr LrLmvqsLr

LmRs sLs d

dt

iqs 2

wherec is the flux linkage; L is the inductance;v is the voltage; R is the resistance; iis the

current and s 1L2m=LrLs is the motor leakage coefficient. The subscripts r and sdenote respectively the rotor and stator values referred to the stator, and the subscripts d

andq denote the dq-axis components in the stationary reference frame.

The rotor flux equations in the stationary frame are [3]

_cdr Lm

Tridswrcqr

1

Trcdr 3

_cqr Lm

Triqswrcdr

1

Trcqr 4

where wr is the rotor electrical speed and Tr=Lr/Rr is the rotor time constant.


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The angleye of the rotor flux vector (cr ) in relation to the d-axis of the stationary frame

is defined as follows:

ye arctancqr

cdr 5

being its derivative:

_ye we cdr

_cqrcqr_cdr

c2dr c2qr

6

Substituting Eqs. (3) and (4) in Eq. (6) it is obtained:

we wrLm

Tr

cdriqscqrids

c2dr c2qr

! 7

Then, substituting Eq. (6) in Eq. (7), and solving for wr we obtain

wr 1

c2rcdr

_cqrcqr_cdr

Lm

Trcdriqscqrids

8

wherec2r c2dr c

2qr.

Therefore, given a complete knowledge of the motor parameters, the instantaneous

speedwrcan be calculated from the previous equation, where the stator measured current

and voltages, and the rotor flux estimation obtained from a rotor flux observer based on

Eqs. (1) and (2) have been employed.

3. Variable structure robust speed control with adaptive sliding gain

In general, the mechanical equation of an induction motor can be written as

J _wmBwmTL Te 9

where JandBare the inertia constant and the viscous friction coefficient of the induction

motor system respectively; TL is the external load; wm is the rotor mechanical speed in

angular frequency, which is related to the rotor electrical speed by wm=2wr/pwherep is the

pole numbers, and Te denotes the generated torque of an induction motor, defined as [3]

Te 3p4

LmLr

cedrieqsc

eqri

eds 10

where cedr andceqr are the rotor-flux linkages, the subscript e denotes that the quantity is

referred to the synchronously rotating reference frame; iqse and ids

e are the stator currents,

andp is the pole number.

The relation between the synchronously rotating reference frame and the stationary

reference frame is computed by the so-called reverse Parks transformation:

xa

xb

xc264 375

cosye sinye

cosye2p=3 sinye2p=3

cosye2p=3 sinye2p=3264 375 xd

xq" # 11

where ye is the angle position between the d-axis of the synchronously rotating and the

stationary reference frames, and the quantities are assumed to be balanced.


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Using the field-orientation control principle [3] the current component idse is aligned in

the direction of the rotor flux vector cr, and the current component iqse is aligned in the

direction perpendicular to it. Under these conditions, it is satisfied that

ceqr 0; c

edr jcrj 12

Fig. 1shows the vectorial diagram of the induction motor in the stationary and in the

synchronously rotating reference frames. The subscripts s indicates the stationary frame

and the subscript e indicates the synchronously rotating reference frame.

Therefore, taking into account the previous results, the equation of induction motor

torque (10) is simplified to

Te 3p

4

Lm

Lrcedri

eqs KTi

eqs 13

where the torque constant, KT, is defined as follows:

KT 3p

4

Lm

Lrce

dr 14

ce

dr being the command rotor flux.

With the above-mentioned field orientation, the dynamics of the rotor flux is given by[3]

dcedrdt

cedr

Tr

Lm

Trieds 15

Then, the mechanical equation (9) becomes

_wmawmf bieqs 16

Fig. 1. Vectorial diagram of the induction motor.


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where the parameters are defined as

aB

J; b

KT

J ; f

TL

J 17

Now, we are going to consider the previous mechanical equation (16) with uncertaintiesas follows:

_wm a Dawmf Df b Dbieqs 18

where the terms Da, Db and Df represent the uncertainties of the terms a, b and f

respectively. It should be noted that these uncertainties are unknown, and that the precise

calculation of an upper bound is, in general, rather difficult to achieve.

Let us define the tracking speed error as follows:

et wmtwmt 19

wherewmn

is the rotor speed command.Taking the derivative of the previous equation with respect to time yields

_et _wm _wm aet ut dt 20

where the following terms have been collected in the signal u(t):

ut bieqstawmtft _w

mt 21

and the uncertainty terms have been collected in the signal d(t),

dt DawmtDft Dbieqst 22

To compensate for the above described uncertainties present in the system, a sliding

adaptive control scheme is proposed. In the sliding control theory, the switching gain must

be constructed so as to attain the sliding condition[20,30]. In order to meet this condition a

suitable choice of the sliding gain should be made to compensate for the uncertainties. To

select the sliding gain vector, an upper bound of the parameter variations, unmodelled

dynamics, noise magnitudes, etc. should be given, but in practical applications there are

situations in which these bounds are unknown, or at least difficult to calculate. A solution

could be to choose a sufficiently high value for the sliding gain, but this approach could

cause a too high control signal, or at least more control activity than needed in order to

achieve the control objective.

One possible way to overcome this difficulty is to estimate the gain and to update it by

means of some adaptation law, so that the sliding condition is achieved.

Now, we are going to propose the sliding variable S(t) with an integral component as

St et

Z t0

aketdt 23

wherek is a constant gain, and a is a parameter that was already defined in Eq. (17).

Then the sliding surface is defined as

St et Z t

0

aketdt 0 24

Now, we are going to design a variable structure speed controller, that incorporates an

adaptive sliding gain, in order to control the AC motor drive

ut ketbtgsgnS 25


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where thekis the gain defined previously, b is the estimated switching gain, g is a positive

constant,S is the sliding variable defined in Eq. (23) and sgn is the sign function.

The switching gain b is adapted according to the following updating law:

_b gjSj; b0 0 26

whereg is a positive constant that let us choose the adaptation speed for the sliding gain.

In order to obtain the speed trajectory tracking, the following assumptions should be

formulated:

A1 The gain kmust be chosen so that the term (ak) is strictly positive. Therefore the

constant kshould be k4a.

A2 There exits an unknown finite non-negative switching gain b such that

b4dmax Z; Z40

wheredmaxZjdtj 8t andZ is a positive constant.

Note that this condition only implies that the uncertainties of the system are bounded

magnitudes.

A3 The constant g must be chosen so that gZ1.

Theorem 1. Consider the induction motor given by Eq.(18).Then,if assumptionsA1A3

are verified, the control law (25) leads the rotor mechanical speed wm(t) so that the speed

tracking error e(t)=wm(t)wmn (t) tends to zero as the time tends to infinity.

The proof of this theorem will be carried out using the Lyapunov stability theory.

Proof. Define the Lyapunov function candidate:

Vt 1

2StSt

1

2~bt ~bt 27

where S(t) is the sliding variable defined previously and ~bt btb

Its time derivative is calculated as

_Vt St _St ~bt_~b t

S_e ake ~bt_b t

Saeud keae ~bgjSj

Sud ke bbgjSj

SkebgsgnS d ke bbgjSj

SdbgsgnS bgjSjbgjSj

dSbgjSj bgjSjbgjSj 28

rjdjjSjbgjSjrjdjjSjdmax ZgjSj

jdjjSjdmaxgjSjZgjSj

rZgjSj 29


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then

_Vtr0 30

It should be noted that Eqs. (23), (20), (25) and (26), and the assumptions A2andA3

have been used in the proof. &

Using Lyapunovs direct method, since V(t) is clearly positive-definite, _Vt is negative

semidefinite andV(t) tends to infinity as S(t) and ~bttends to infinity, then the equilibrium

at the origin St; ~bt 0; 0 is globally stable, and therefore the variables S(t) and ~btare bounded. Then, since S(t) is bounded one has that e(t) is also bounded.

Besides, computing the derivative of Eq. (23), it is obtained:

_St _et aket 31

then, substituting Eq. (20) in Eq. (31),

_St aet ut dt aket

ket dt ut 32

From Eq. (32) we can conclude that _St is bounded because e(t), u(t) and d(t) are

bounded.

Now, from Eq. (28) it is deduced that

Vt d _Stbgd

dtjStj 33

which is a bounded quantity because _St is bounded.Under these conditions, since V is bounded, _V is a uniformly continuous function, so

Barbalats lemma let us conclude that _V-0 as t-1, which implies thatSt-0 as t-1.

Therefore S(t) tends to zero as the time t tends to infinity. Moreover, all trajectories

starting off the sliding surfaceS=0 must reach it asymptotically and then will remain on this

surface. This systems behavior, once on the sliding surface is usually calledsliding mode[20].

When the sliding mode occurs on the sliding surface (24), then St _St 0, and

therefore the dynamic behavior of the tracking problem (20) is equivalently governed by

the following equation:

_

St 0 ) _et aket 34Then, under assumption A1, the tracking error e(t) converges to zero exponentially.

It should be noted that, a typical motion under sliding-mode control consists of a reaching

phase during which trajectories starting off the sliding surface S=0 move towards it and

reach it, followed by asliding phase during which the motion is confined to this surface and

where the system tracking error, represented by the reduced-order model (34), tends to zero.

Finally, the torque current command,iqsen(t), can be obtained directly substituting Eq. (25)

in Eq. (21):

ieqs t 1

b

kebgsgnS awm _wmf 35

Therefore, the proposed variable structure speed control with adaptive sliding gain

resolves the speed tracking problem for the induction motor, with uncertainties in

mechanical parameters and load torque.


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4. Simulation results

In this section we will study the speed regulation performance of the proposed adaptive

sliding-mode field oriented control versus speed reference and load torque variations by

means of simulation examples. In particular, the example presented consist of a repre-sentative speed reference tracking problem, combined with load torque variations during

the evolution of the experiment and considering a certain degree of uncertainty, in order to

attain a complete scope of the behavior of the system.

The block diagram of the proposed robust control scheme is presented in Fig. 2.

The block VSC controller represents the proposed adaptive sliding-mode controller, and

it is implemented by Eqs. (23), (35), and (26). The block limiter limits the current applied

to the motor windings so that it remains within the limit value, and it is implemented by a

saturation function. The block dqe-abc makes the conversion between the synchro-

nously rotating and stationary reference frames, and is implemented by Eq. (11). The block

current controller consists of a three hysteresis-band current PWM control, which is

basically an instantaneous feedback current control method of PWM where the actual

current (iabc) continually tracks the command current (iabcn ) within a hysteresis band. The

block PWM inverter is a six IGBT-diode bridge inverter with 780 V DC voltage source.

The block field weakening gives the flux command based on rotor speed, so that the PWM

controller does not saturate. The block idsen calculation provides the current reference ids

en

from the rotor flux reference through Eq. (15). The block wr estimator represents the

proposed rotor speed and synchronous speed estimator, and it is implemented by Eqs. (8)

and (6) respectively. Finally, the block IM represents the induction motor.

The induction motor used in this case study is a 50 HP, 460 V, four pole, 60 Hz motorhaving the following parameters:Rs 0:087 O,Rr 0:228 O,Ls=35.5 mH,Lr=35.5 mH,andLm=34.7 mH.

The system has the following mechanical parameters: J=1.357 kg m2 and B=0.05 N

m s. It is assumed that there is an uncertainty around 20% in the system parameters, which

will be overcome by the proposed sliding control.

The following values have been chosen for the controller parameters: k=45 andg 30.

In this example the motor starts from a standstill state and we want the rotor speed to

follow a speed command that starts from zero and accelerates until the rotor speed is

Fig. 2. Block diagram of the proposed adaptive sliding-mode control.


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100 rad/s, then the rotor speed is maintained constant and after, at time 1.3 s, the rotor

decelerates until the rotor speed is 80 rad/s. In this simulation example, the system starts

with an initial load torque TL=0 N m and at time t=2.3 s the load torque steps from

TL=0 to 200 N m, and as before, it is assumed that there is an uncertainty around 20% in

the load torque.Fig. 3shows the desired rotor speed (dashed line) and the real rotor speed (solid line).

As it may be observed, after a transitory time in which the sliding gain is adapted, the rotor

speed tracks the desired speed in spite of system uncertainties. However, at time t=2.3 s a

small speed error can be observed. This error appears because a torque increment occurs at

this time, so that the control system loses the so-called sliding mode because the actual

sliding gain is too small in order to overcome the new uncertainty introduced in the system

due to the new torque. But then, after a short time the sliding gain is adapted once again so

that this gain can compensate for the system uncertainties which eliminates the rotor

speed error.

Fig. 4presents the time evolution of the estimated sliding gain. The sliding gain starts

from zero and then it is increased until its value is high enough to compensate for the

system uncertainties. Besides, the sliding remains constant because the system uncertainties

remain constant as well. Later, at time 2.3 s, there is an increment in the system

uncertainties caused by the increment in the load torque. Therefore, the sliding should be

adapted once again in order to overcome the new system uncertainties. As it can be seen in

the figure after the sliding gain is adapted, it remains constant again, since the system

uncertainties remain constant as well.

It should be noted that the adaptive sliding gain allows to employ a smaller sliding gain,

so that the value of the sliding gain do not have to be chosen high enough to compensatefor all possible system uncertainties, because with the proposed adaptive scheme the sliding

gain is adapted (if it is necessary) when a new uncertainty appears in the system in order to

surmount this uncertainty.

Fig. 5shows the time evolution of the sliding variable. In this figure it can be seen that

the system reach the sliding condition (S(t)=0) at timet=0.13 s, but then the system loses

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Time (s)

RotorSpeed(rad/s)

wm*

wm

Fig. 3. Reference and real rotor speed signals (wmn , wm).


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this condition at time t=2.3 s due to the torque increment which, in turn, produces an

increment in the system uncertainties that cannot be compensated by the actual value of

the sliding gain. However, after a transitory time in which the sliding gain is adapted in

order to compensate the new system uncertainty, the system reaches the sliding

condition again.

Fig. 6shows the current of one stator winding. This figure shows that in the initial state,

the current signal presents a high value because a high torque is necessary to increment therotor speed due to the rotor inertia. Then, in the constant-speed region, the motor torque

only has to compensate the friction and therefore, the current is lower. Finally, at time

t=2.3 s the current increases because the load torque has been increased.

0 0.5 1 1.5 2 2.5 30.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

SlidingVariable

Fig. 5. Sliding variable.

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

Time (s)

SlidingGain

Fig. 4. Estimated sliding gainb.


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Fig. 7shows the motor torque. As in the case of the current (Fig. 6), the motor torque

has a high initial value in the speed acceleration zone and then the value decreases in a

constant region. Later, at time t=1.3 s, the motor torque decreases again in order to

reduce the rotor speed. Finally, at time t=2.3 s the motor torque increases in order to

compensate the load torque increment. In this figure it may be seen that in the motor

torque appears the so-called chattering phenomenon, however this high frequency changes

in the torque will be filtered by the mechanical system inertia.

5. Conclusions

In this paper sensorless robust vector control for induction motor drives with an

adaptive variable sliding-mode vector control law has been presented. The rotor speed

0 0.5 1 1.5 2 2.5 3500

400

300

200

100

0

100200

300

400

500

Time (s)

StatorCurrent

Fig. 6. Stator current (isa).

0 0.5 1 1.5 2 2.5 3

100

50

0

50

100

150

200

250

300

MotorTor

que(N)

Time (s)

Fig. 7. Motor torque (Te).


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estimator is based on stator voltage equations and rotor flux equations in the stationary

reference frame. It is proposed as a variable structure control which uses an integral sliding

surface to relax the requirement of the acceleration signal, that is usual in conventional

sliding-mode speed control techniques. Due to the nature of the sliding control this control

scheme is robust under uncertainties caused by parameter errors or by changes in the loadtorque. Moreover, the proposed variable structure control incorporates and adaptive

algorithm to calculate the sliding gain value. The adaptation of the sliding gain, on the one

hand avoids the need of computing the upper bound of the system uncertainties, and on

the other hand allows to employ as smaller sliding gain as possible to overcome the actual

system uncertainties. Then the control signal of our proposed variable structure control

schemes will be smaller than the control signals of the traditional variable structure control

schemes, because in these traditional schemes the sliding gain value should be chosen high

enough to overcome all the possible uncertainties that could appear in the system along

the time.

The closed loop stability of the design presented in this paper has been proved thought

Lyapunov stability theory. Finally, by means of simulation examples, it has been shown

that the proposed control scheme performs reasonably well in practice, and that the speed

tracking objective is achieved under uncertainties in the parameters and load torque.

Acknowledgments

The authors are very grateful to the Basque Government by the support of this work

through the project S-PE09UN12 and to the UPV/EHU by its support through project

GUI07/08.

References

[1] W. Leonhard, Control of Electrical Drives, Springer, Berlin, 1996.

[2] P. Vas, Vector Control of AC Machines, Oxford Science Publications, Oxford, 1994.

[3] B.K. Bose, Modern Power Electronics and AC Drives, Prentice Hall, New Jersey, 2001.

[4] R. Beguenane, M.A. Ouhrouche, A.M. Trzynadlowski, A new scheme for sensorless induction motor control

drives operating in low speed region, Mathematics and Computers in Simulation 71 (2006) 109120.

[5] S. Sunter, Slip energy recovery of a rotor-side field oriented controlled wound rotor induction motor fed by

matrix converter, Journal of the Franklin Institute 345 (2008) 419435.

[6] M. Comanescu, An induction-motor speed estimator based on integral sliding-mode current control, IEEETransactions on Industrial Electronics 56 (9) (2009) 34143423.

[7] M.I. Marei, M.F. Shaaban, A.A. El-Sattar, A speed estimation unit for induction motors based on adaptive

linear combiner, Energy Conversion and Management 50 (2009) 16641670.

[8] A.Y. Alanis, E.N. Sanchez, A.G. Loukianov, E.A. Hernandez, Discrete-time recurrent high order neural

networks for nonlinear identification, Journal of the Franklin Institute 347 (2010) 12531265.

[9] T-J. Ren, T-C. Chen, Robust speed-controlled induction motor drive based on recurrent neural network,

Electric Power System Research 76 (2006) 10641074.

[10] M. Montanari, S. Peresada, A. Tilli, A speed-sensorless indirect field-oriented control for induction motors

based on high gain speed estimation, Automatica 42 (2006) 16371650.

[11] R. Marino, P. Tomei, C.M. Verrelli, An adaptive tracking control from current measurements for induction

motors with uncertain load torque and rotor resistance, Automatica 44 (2008) 25932599.

[12] J.B. Oliveira, A.D. Araujo, S.M. Dias, Controlling the speed of a three-phase induction motor using asimplified indirect adaptive sliding mode scheme, Control Engineering Practice 18 (2010) 577584.

[13] M.A. Fnaiech, F. Betin, G.A. Capolino, F. Fnaiech, Fuzzy logic and sliding-mode controls applied to six-

phase induction machine with open phases, IEEE Transactions on Industrial Electronics 57 (1) (2010)

354364.


8/10/2019 1-s2.0-S001600321000270X-main

15/15

[14] M. Montazeri-Gh, A. Poursamad, B. Ghalichi, Application of genetic algorithm for optimization of control

strategy in parallel hybrid electric vehicles, Journal of the Franklin Institute 343 (2006) 420435.

[15] A. Benchaib, C. Edwards, Nonlinear sliding mode control of an induction motor, International Journal of

Adaptive Control and Signal Procesing 14 (2000) 201221.

[16] O. Barambones, A.J. Garrido, A sensorless variable structure control of induction motor drives, Electric

Power Systems Research 72 (2004) 2132.

[17] R. Yazdanpanah, J. Soltani, G.R. Arab Markadeh, Nonlinear torque and stator flux controller for induction

motor drive based on adaptive inputoutput feedback linearization and sliding mode control, Energy

Conversion and Management 49 (2008) 541550.

[18] B. Castillo-Toledo, S. Di Gennaro, A.G. Loukianov, J. Rivera, Discrete time sliding mode control with

application to induction motors, Automatica 44 (2008) 30363045.

[19] T. Or"owska-Kowalska, M. Kaminski, K. Szabat, Implementation of a sliding-mode controller with an

integral function and fuzzy gain value for the electrical drive with an elastic joint, IEEE Transactions on

Industrial Electronics 57 (4) (2010) 13091317.

[20] V.I. Utkin, Sliding mode control design principles and applications to electric drives, IEEE Transactions on

Industrial Electronics 40 (1993) 2636.

[21] H. Yang, Y. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via deltaoperator approach, Journal of the Franklin Institute 347 (2010) 11991213.

[22] I. Boiko, Frequency domain precision analysis and design of sliding mode observers, Journal of the Franklin

Institute 347 (2010) 899909.

[23] T. Furuhashi, S. Sangwongwanich, S. Okuma, A position-and-velocity sensorless control for brushless DC

motors using an adaptive sliding mode observer, IEEE Transactions on Industrial Electronics 39 (1992)

8995.

[24] A.B. Proca, A. Keyhani, J.M. Miller, Sensorless sliding-mode control of induction motors using operating

condition dependent models, IEEE Transactions on Energy Conversion 18 (2003) 205212.

[25] G. Bartolini, A. Pisano, E. Punta, E. Usai, A survey of applications of second-order sliding mode control to

mechanical systems, International Journal of Control 76 (2003) 875892.

[26] M. Rashed, K.B. Goh, M.W. Dunnigan, P.F.A. MacConnell, A.F. Stronach, B.W. Williams, Sensorless

second-order sliding-mode speed control of a voltage-fed induction-motor drive using nonlinear statefeedback, IEE Proceedings Electric Power Applications 152 (2005) 11271136.

[27] C. Aurora, A. Ferrara, A sliding mode observer for sensorless induction motor speed regulation,

International Journal of Systems Science 38 (2007) 913929.

[28] Y. Xia, Z. Zhu, C. Li, H. Yang, Q. Zhu, Robust adaptive sliding mode control for uncertain discrete-time

systems with time delay, Journal of the Franklin Institute 347 (1) (2010) 339357.

[29] M.C. Pai, Design of adaptive sliding mode controller for robust tracking and model following, Journal of the

Franklin Institute 347 (2010) 18381849.

[30] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1991.


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