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Electric Power Systems Research 77 (2007) 604–618

Adaptive sliding mode observer for induction motorusing two-time-scale approach

A. Mezouar ∗, M.K. Fellah, S. HadjeriIntelligent Control and Electrical Power Systems Laboratory, Department of Electrical Engineering, Faculty of Sciences Engineering,

Djillali Liabes University, (22 000) Sidi Bel Abbes, Algeria

Received 29 July 2005; received in revised form 5 April 2006; accepted 25 May 2006Available online 14 July 2006

bstract

This paper presents an original method for the design of a robust adaptive sliding mode current and flux observer for induction motor drivesing two-time-scale approach. This approach, based on the singular perturbation theory, decomposes the original system of the observer errorynamics into separate slow and fast subsystems of lower dimensions and permits a simple design and sequential determination of the observerains. For the proposed observer, the rotor speed signal is assumed to be available. The stator currents and rotor flux are observed on the stationaryeference frame using sliding mode concept, and the adaptive rotor time-constant is derived from Lyapunov stability theory using measured and

stimated currents and estimated rotor flux. The control algorithm is based on the indirect field-oriented sliding mode control to keep the machineeld oriented. The control-observer scheme seeks to provide asymptotic tracking of speed and rotor flux in spite of the presence of an uncertain

oad torque and unknown value of the rotor resistance. The effectiveness of this control algorithm has been successfully verified through computerimulations.

2006 Elsevier B.V. All rights reserved.

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eywords: Induction motor drives; Singular perturbation theory; Sliding mode

. Introduction

In the past few decades, the control of the induction motoronstitutes a theoretically challenging problem since the dynam-cal system is non-linear, a part of state variables are not easilyeasurable and the physical parameters are most often impre-

isely known [1,2]. One of the most significant developmentsn this area was the field-oriented control. The orientation ofhe rotor flux made possible to act independently on the rotorux and the electromagnetic torque through the intermediary of

he components of the stator voltage [1,2]. Unfortunately, thisontrol approach suffers from sensitivity to the motor parameterariations. Indeed, there are two ways to solve this problem. Therst is to perform on-line identification of the motor parameters

nd accordingly update the values used in the controller. Thether solution is to use a robust control algorithm insensitive tohe motor parameter variations.

∗ Corresponding author. Tel.: +21 37 232 0609; fax: +21 34 847 4262.E-mail addresses: a mezouar@yahoo.com, mezouar@univ-sba.dz

A. Mezouar).

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378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2006.05.010

ol; Sliding mode observer

In the last years, the sliding mode technique has beenidely studied and developed for the control and state esti-ation problems since the works of Utkin [3]. This control

echnique allows good steady state and good dynamic behav-or in the presence of system parameters variation and distur-ances [4–6]. Several methods of applying sliding mode controlo induction motor drives have been presented [4–8]. All ofhese methods have a common feature: the analysis and designf the sliding mode controller are based on the mathemati-al model of the induction motor as used in indirect vectorontrol.

Most control schemes for induction motor require accuratenformation on state variables and parameters. Indeed, fluxes aresually estimated with measured stator currents, stator voltages,nd motor speed by using the current model for rotor flux or volt-ge model for stator flux [9–13]. However, since the estimateduxes are basically dependent on the motor model, variation

n parameters inevitably propagates to flux estimation. Among

hese parameters, rotor resistance which varies mainly depend-ng on temperature.

One of the solutions to reduce this type of estimation erroro produce better control performance is to design an on-line

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dentification scheme for parameter adaptation compensatingor the parameter variation. Recently, the estimation of theotor resistance or rotor time-constant has constituted a subjectf interest for many research teams, and several solutions forstimating the state variables and parameters of the inductionotor have been proposed in the literature such as: the extendeduenberger observer (ELO) [14,15], the extended Kalman fil-

er (EKF) [15–17], model reference adaptive systems (MRAS)18–20], adaptive observer (AO) [21,22], sliding mode observerSMO), [23,24] and newly fuzzy logic and neural networksbservers [25].

On the other hand, singular perturbation theory provides theean to decompose two-time-scale systems into slow and fast

ubsystems of lower order dimension described in separate time-cales, which greatly simplify their structural analysis and anyubsequent control design. Then, the control (and/or observer)esign may be done for each lower order subsystem, and theombined results yield to a composite control (and/or observer)or the original system [26–28].

So, the idea of combining singular perturbation theorynd sliding mode technique constitutes a good possibilityo achieve classical control objectives for systems havingnmodeled or parasitic dynamics and parametric uncertainties29–31].

In this paper, an adaptive sliding mode observer is developedor the simultaneous estimation of the rotor flux componentsnd of the rotor time-constant for induction motor under thessumption that only the stator currents and the motor speedre available for measurement. Singular perturbation theory issed for having a sequential and simple design of this observer,nd the flux observer stability is insured through the Lyapunovheory.

This paper is organized as follows: the main results ofhe two-time-scales approach and the design of the generalwo-time-scales sliding mode observer are presented in Sec-ions 2 and 3, respectively. In Section 4, we briefly reviewhe indirect field-oriented sliding mode control of induction

otors. The design of a two-time-scale sliding-mode observeror the induction motor model is presented in Section 5. Inhat section, a study of stability analysis of this observer is

ade via singular perturbation method with sliding mode con-ept and Lyapunov stability theory. In Section 6, and throughimulation, the studied observer is associated to the indi-ect field-oriented sliding-mode control where rotor fluxesnd rotor resistance are replaced by those delivered by thebserver. Finally, in Section 7, we give some comments andonclusions.

. Two-time-scale approach

The two-time-scale approach, based on singular perturbationheory, can be applied to systems where the state variables cane split into two sets, one having “fast” dynamics, the other

aving “slow” dynamics. The difference between the two sets ofynamics can be distinguished by the use of a small multiplyingcalar ε. Generally, the scaling parameter ε is the speed ratio ofhe slow versus fast phenomena.

ms Research 77 (2007) 604–618 605

If the slow states are expressed in the t time-scale, then, theast ones will be in the τ time-scale defined by

= t − t0

ε(1)

here t0 is the initial time.The reader is referred to [26] and [27] for the general theory

n singular perturbation.

.1. Non-linear singularly perturbed systems

Let us consider the following class of non-linear singularlyerturbed systems described by the so-called standard singularlyerturbed form

d

dtx = fx(x, z, u, t, ε), x(t0) = x0

εd

dtz = fz(x, z, u, t, ε), z(t0) = z0

(2)

here x ∈ �n is the slow state, z ∈ �m is the fast state, u ∈ �p

s the control input and ε is a small positive parameter suchhat ε ∈ [0, 1]. fx and fz are assumed to be bounded and analyticeal vector fields, and consider a vector of measurement that isinearly related to the fast state vector as

= z, y ∈ �m (3)

.2. Slow reduced subsystem

In the limiting case, as ε → 0 in (2), the asymptotically stableast transient decays ‘instantaneously’, leaving the reduced-rder model in the t time-scale defined by the quasi-steady-statess(t) and zs(t).

d

dtxs = fx(xs, zs, us, t, 0) (4)

= fz(xs, zs, us, t, 0) (5)

and the substitution of a root of (5)

s = h(xs, us, t) (6)

into (4) yields a reduced model

d

dtxs = fx(xs, h(xs, us, t), us, t, 0), xs(t0) = x0 (7)

here the index (s) indicates that the associated quantity belongso the system without ε.

.3. Fast reduced subsystem

The fast dynamic (also known as boundary layer system)enoted zf, which represents the derivation of zs from z isbtained by transforming the slow time scale t to the fast time-cale τ = (t − t0)/ε. System of Eq. (2) becomes

d

dτx = εfx(x, z, u, ετ + t0)

d

dτz = fz(x, z, u, ετ + t0)

(8)

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Introducing the derivation of zs from z, i.e., zf = z − zs andgain examine the limit as ε → 0. Then, it yields

d

dτzf = fz(x0, zs(0) + zf(τ), uf(τ), t0) (9)

ith

f(0) = z0 − zs(0)

here uf = u − us is the fast part of the input control.

.4. Two-time-scale states approximation

Fast and slow variables given by (7) and (9) can be combinednto a composite structure in order to approximate the originaltates of (2) as given in [26]:

x = xs(t) + O(ε)

z = zs(t) + zf(τ) + O(ε)(10)

. Two-time-scale sliding mode observer

Consider the above continuous non-linear singularly per-urbed system of Eq. (2) which is given by

x = fx(x, z, u, ε)

εz = fz(x, z, u, ε)(11)

In addition, it is assumed that the above system is observable.onsequently, the observer design may be considered for the

tate observation of slowly variables from the measurement ofast variables [29–31].

.1. Sliding mode observer design

By structure, observer based on sliding mode approach is veryimilar to the standard full order observer with replacement ofhe linear corrective terms by a discontinuous function [8].

The corresponding sliding mode observer for the system of11) can be written as a replica of the system with an additionalon-linear auxiliary input term as follows:

˙x = fx(x, z, u, ε) + GxIs

ε ˙z = fz(x, z, u, ε) + GzIs(12)

here Is = sign(S(y, y)) is the switching function. Gx and Gz

re the observer gains with (n × m) and (m × m) dimensions,espectively, to be determined.

The sliding surface S can be chosen as a linear function ofy − y) as given in [8], so

(y, y) = �(y − y) (13)

here � is a (m × m) invariant matrix to be specified, and

y − y)T = [ y1 − y1) (y2 − y2) . . . (ym − ym) ] I

ms Research 77 (2007) 604–618

The error dynamics is calculated by subtracting (12) from11):

ex = fx(x, z, u, ε) − fx(x, z, u, ε) − GxIs

εez = fz(x, z, u, ε) − fz(x, z, u, ε) − GzIs(14)

r

ex = �fx − GxIs

εez = �fz − GzIs(15)

here

ex = x − x, ez = z − z,

�fx = fx(x, z, u, ε) − fx(x, z, u, ε),

�fz = fz(x, z, u, ε) − fz(x, z, u, ε).

The observer gains design can be based on sequential appli-ation of resulted subsystems of (15) by applying singularerturbation methodology. We first need to analyze the fastariables tracking-properly using the so-called reaching con-ition (based on measurable state variables), and, thereafter, thelow variables asymptotic-convergence (for inaccessible stateariables).

.2. Stability analysis in the fast time-scale

For fast error dynamic subsystem, the associated time scales defined by τ = (t − t0)/ε, then (15) can be transformed into

dex

dτ= ε(�fx − GxIs)

dez

dτ= �fz − GzIs

(16)

Setting ε = 0 in (16), it yields

dez

dτ= �fz − GzIs (17)

ith

dex

dτ= 0

In this time-scale, the stability analysis consists of determin-ng the observer gain Gz so that in this time scale (τ), the surface(τ) = 0 is attractive.

It can be shown that when sliding mode occurs on S(τ), thequivalent value of the discontinuous observer auxiliary inputs found by solving the Eq. (17) for GzIs after insuring zero forez/dτ such as

zIs = �fz

and the equivalent switching vector is obtained as

˜s = G−1z �fz (18)

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A. Mezouar et al. / Electric Power

.3. Stability analysis in the slow time-scale

Slowly error dynamic subsystem can be found by making= 0 in (15), so

dex

dt= �fx − GxIs (19)

= �fz − GzIs (20)

From (20), the equivalent switching vector can be re-founds

˜s = G−1z �fz

herefore, by appropriate choice of Gx, the desired rate of con-ergence ex → 0 can be obtained.

. Sliding-mode-control review of induction motor

Assuming that the induction model system is controllablend observable, the sliding-mode-control consists of two phases4,5]:

designing an equilibrium surface, called sliding surface, suchthat any state trajectory of the plant restricted to the slidingsurface is characterized by the desired behavior;designing a discontinuous control law to force the system tomove on the sliding surface in a finite time.

.1. Dynamic model of induction machine

Under the assumptions of linearity of the magnetic circuitnd neglecting iron losses, the state space model of three-phasenduction motor expressed in the synchronously rotating refer-nce frame (d − q) is [1]:

d

dtisd = − Rλ

σLsisd + ωsisq + μ

σLs

1

Trφrd

+ μ

σLsωφrq + 1

σLsvsd

d

dtisq = −ωsisd − Rλ

σLsisq − μ

σLsωφrd

+ μ

σLs

1

Trφrq + 1

σLsvsq

d

dtφrd = M

Trisd − 1

Trφrd + ωslφrq

d

dtφrq = M

Trisq − ωslφrd − 1

Trφrq

dω = p

(Tem − TL) − fω

(21)

dt J J

here the state variables are the stator currents (isd, isq), theotor fluxes (φrd, φrq) and the rotor speed ω. Stator voltagesvsd, vsq) and slip frequency ωsl (ωsl = ωs − ω) are the control

4

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ms Research 77 (2007) 604–618 607

ariables. The electromagnetic torque expressed in terms of thetate variables is

em = pM

Lr(φrdisq − φrqisd) (22)

ith constants defined as follows:

λ = Rs + M2

L2r

Rr, σ = 1 − M2

LsLr, μ = M

Lr.

.2. Induction motor model with rotor field oriented

Among the various sliding mode control solutions for induc-ion motor proposed in the literature, the one based on indirecteld orientation can be regarded as the simplest one. Its pur-ose is to directly control the inverter switching by use of twowitching surfaces.

The induction motor equations in the synchronously rotatingeference frame (d − q), oriented in such a way that the rotorux vector points into d-axis direction, are:

d

dtω = f1

d

dtφrd = f2

d

dtisd = f3 + 1

σLsvsd

d

dtisq = f4 + 1

σLsvsq

(23)

nd

d

dtθs = ω + ωsl (24)

ith

sl = M

Tr

isq

φrd(25)

f1 = kcφrdisq − p

JTL − f

Jω

f2 = M

Trisd − 1

Trφrd

f3 = − Rλ

σLsisd + ωsisq + μ

σLs

1

Trφrd

f4 = −ωsisd − Rλ

σLsisq − μ

σLsωφrd

(26)

nd

c = p2M

JLr

.3. Speed and flux sliding mode controller

Using the reduced non-linear induction motor model of Eq.23), it is possible to design both a speed and a flux sliding mode

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ontrollers. Let us define the sliding surfaces

Sc1 = Sc1(ω) = λω(ω∗ − ω) + d

dt(ω∗ − ω)

Sc2 = Sc2(φr) = λφ(φ∗r − φdr) + d

dt(φ∗

r − φdr)

(27)

here λω > 0, λφ > 0. ω* and φ∗r are the reference speed and the

eference rotor flux, respectively.To determine the control law that leads the sliding functions

27) to zero in finite time, one has to consider the dynamics ofc = (Sc1, Sc2)T, described by

˙c = F + DVS (28)

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[vsq

vsd

]

roposition (1). If the Lyapunov theory of stability is used tonsure that Sc is attractive and invariant, the following conditionas to be satisfied

Tc · Sc < 0 (29)

So, it is possible to choose the switching control law for statoroltages as follows

vsq

vsd

]= −D−1F − D−1

[Kω 0

0 Kφ

][sign(Sc1)

sign(Sc2)

](30)

here

Kω > 0

Kφ > 0(31)

roof (1). Let consider the Lyapunov function

= 1

2ST

c Sc (32)

Its time derivative is

˙ = STc Sc (33)

Using Eqs. (28) and (30), we can rewrite (33) as

˙ = STc

(F + D

(−D−1F − D−1

[Kω 0

0 Kφ

]sign(Sc)

))

•

5

t

ms Research 77 (2007) 604–618

r

˙ = −STc

([Kω 0

0 Kφ

]sign(Sc)

)(34)

Then, the manifold Sc is attractive if U < 0, i.e.,

Kω > 0

Kφ > 0�

The sliding mode causes drastic changes of the control vari-bles introducing high frequency disturbances. To reduce thehattering phenomenon, a saturation function sat(Sc) instead ofhe switching one sign(Sc) has been introduced

at(Sci) =⎧⎨⎩

Sci

δi

if |(Sci)| ≤ δi

sign(Sci) if |(Sci)| > δi

(35)

here δi > 0 and i = 1, 2

emark (1). From the above control law of Eq. (30), it cane see that the implementation of these algorithms requires theoad torque and rotor flux estimations since stator currents, statoroltages and speed rotor are available by measures. In the nextection, we are interested by a robust estimation of rotor flux.he estimated load torque can be easily obtained by using theechanical equation of the motor model with estimated rotoruxes and measured stator currents.

In the following, we will assume to operate with constanteference speed, constant reference rotor flux and constant loadorque, so that ω∗ = 0, φ∗

r = 0 and TL = 0.

. Two-time-scale sliding mode observer design fornduction motor

Consider only the first four equations of the induction motorodel of Eq. (21) in which the speed motor will be considered astime-varying parameter. The objective of the studied observer

s to estimate the unmeasured rotor fluxes and the rotor time-onstant.

The sliding-mode observer design procedure consists of per-orming the following two steps [7]:

Designing of an equilibrium surface such that the estimationerror trajectories restricted to this surface, have desired stabledynamics.Determining of the observer gains to drive the estimation errortrajectories to the sliding surface and maintain it on the set.

.1. Dynamic model of induction motor

Using the model of Eq. (21), the state space model of induc-ion motor, without mechanical equation, expressed in the fixed

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A. Mezouar et al. / Electric Power

tator reference frame (α, β) is

d

dtisα = − Rλ

σLsisα + μ

σLs

1

Trφrα + μ

σLsωφrβ + 1

σLsvsα

d

dtisβ = − Rλ

σLsisβ − μ

σLsωφrα + μ

σLs

1

Trφrβ + 1

σLsvsβ

d

dtφrα = M

Trisα − 1

Trφrα − ωφrβ

d

dtφrβ = M

Trisβ + ωφrα − 1

Trφrβ

(36)

Voltage, current and flux transformation from the syn-hronous to the stationary reference frame and vice versa isade by the rotational transformation as [1,2]:

xd

xq

)= R(θs)

(xα

xβ

)(37)

ith

(θs) =(

cos(θs) sin(θs)

− sin(θs) cos(θs)

), [R(θs)]

−1 = R(−θs) (38)

here x = v, i, φ and θs is the angular displacement of the syn-hronous rotating reference frame.

.2. Singularly perturbed induction motor model

Based on the well-known of the induction machine modelynamics [29–31], the slow variables are (φrα, φrβ) and the fastariables are (isα, isβ). Therefore, the system of Eq. (36) becomes

σLsd

dtisα = −Rλisα + μαrφrα + μωφrβ + vsα

σLsd

dtisβ = −Rλisβ − μωφrα + μαrφrβ + vsβ

d

dtφrα = Mαrisα − αrφrα − ωφrβ

d

dtφrβ = Mαrisβ + ωφrα − αrφrβ

(39)

here

r = Rr

Lr, Rλ = Rs + Mμαr

For which, the corresponding standard singularly perturbedorm with ε = σ Ls, x = (φrα, φrβ)T and z = (isα, isβ)T is

εz1 = −Rλz1 + μαrx1 + μωx2 + vsα

εz2 = −Rλz2 − μωx1 + μαrx2 + vsβ

x1 = Mαrz1 − αrx1 − ωx2(40)

x2 = Mαrz2 + ωx1 − αrx2

emark (2). All parameters of induction motor will be con-idered as constant except rotor resistance. The rotor resistancer will be treated as uncertain parameter with Rrn as its nominal

taat

ms Research 77 (2007) 604–618 609

alue. An addition assumption is that Rr varies slowly (practicalssumption), so that Rr ≈ 0.

The motor speed will be treated as a bounded time-varyingariable.

.3. Singularly perturbed sliding mode observer

From Section 3, the observer equations of the above modelased on the sliding mode concept are the following

ε ˙z1 = −Rλz1 + μαrx1 + μωx2 + vsα + Gz1Is

ε ˙z2 = −Rλz2 − μωx1 + μαrx2 + vsβ + Gz2Is

˙x1 = Mαrz1 − αrx1 − ωx2 + Gx2Is

˙x2 = Mαrz2 + ωx1 − αrx2 + Gx2Is

(41)

here αr = αr + �αr and Rλ = Rs + Mμαr, in which

ˆ r = Rr

Lr= Rr

Lr+ �Rr

Lr(42)

here xi and zj are the estimation of xi and zj for i ∈ {1, 2} and∈ {1, 2}. Gx1 , Gx2 ,Gz1 and Gz2 are the observer gains.

The switching vector Is is chosen as

s =[

sign(s1)

sign(s2)

](43)

ith

=[

s1

s2

]=[

z1 − z1

z2 − z2

]=[

ez1

ez2

](44)

Setting exi = xi − xi and ezj = zj − zj for i ∈ {1, 2} and∈ {1, 2}, and using Eqs. (40) and (41), the estimation errorynamics are

εez1 = μ[(+αrex1 + ωex2 ) + �αr(Mz1 − x1)] − Gz1Is

εez2 = μ[(−ωex1 + αrex2 ) + �αr(Mz2 − x2)] − Gz2Is

ex1 = −[(+αrex1 + ωex2 ) + �αr(Mz1 − x1)] − Gx2Is

ex2 = −[(−ωex1 + αrex2 ) + �αr(Mz2 − x2)] − Gx2Is(45)

Eq. (45) can be expressed in the matrix form as

εez = μ [(αrI2 − ωJ2)ex + �αr(Mz − x)] − GzIs

ex = − [(αrI2 − ωJ2)ex + �αr(Mz − x)] − GxIs(46)

here I2 is the (2 × 2) identity matrix and J2 is the (2 × 2) skewymmetric matrix defined by

2 =[

0 −1

1 0

]

Exploiting the time-properties of the multi-time-scales sys-

em of Eqs. (40) and (41), then ez = (ez1 , ez2 )T are fast variablesnd ex = (ex1 , ex2 )T are slow variables. Therefore, the stabilitynalysis of the system (45) consists of determining Gz1 and Gz2

o ensure the attractiveness of the sliding surface S(τ) = 0 in the

6 Syste

fti

5

tf⎧⎪⎪⎨⎪⎪⎩

m

Pts

G

b

S

PV

S

o

S

Tm

b

S

Rsbr

5o

u

0

e

5d

Pdmci

G

w

P

0

e

w

W

w

10 A. Mezouar et al. / Electric Power

ast time scale. Thereafter, Gx1 and Gx2 are determined, suchhat the reduced-order system obtained when S(τ) ∼= S(τ) ∼= 0s locally stable.

.4. Fast reduced-order error dynamics (current observer)

From singular perturbation theory, the fast reduced-order sys-em of the observation errors can be obtained by introducing theast time-scale τ = (t − t0)/ε. System of Eq. (46) gives

d

dτez = μ [(αrI2 − ωJ2)ex + �αr(Mz − x)] − GzIs

d

dτex = −ε [(αrI2 − ωJ2)ex + �αr(Mz − x)] − εGxIs

(47)

Making ε = 0 in the above system, it yields

d

dτez = μ[(αrI2 − ωJ2)ex + �αr(Mz − x)] − GzIs (48)

d

dτex = 0 (49)

By appropriate choice of the observer gain matrix Gz, slidingode occurs in (48) along the manifold ez = 0.

roposition (2). Assume that ex1 and ex2 are bounded in thisime (practical assumption) and ω varies slowly, and considerystem (48) with the following observer gain matrix

z =[

η1 0

0 η2

]; (50)

The attractivity condition of the sliding surface S(τ) = 0 giveny

T(

dS

dτ

)< 0 (51)

is verified with the set defined by the following inequalities

η1 >∣∣μ [αrex1 + ωex2 + �αr(Mz1 − x1)

]∣∣η2 >

∣∣μ [αrex2 − ωex1 + �αr(Mz2 − x2)]∣∣ (52)

roof (2). Let we use the definite positive Lyapunov function= STS/2. In this time-scale, the derivative of V is given by

T d

dτS = ST[μ[(αrI2 − ωJ2)ex + �αr(Mz − x)] − GzIs]

(53)

r

T d

dτS = −s1[η1sign(s1) − μ[αrex1 + ωex2

+ �αr(Mz1 − x1)]] − s2[η2sign(s2) − μ[αrex2

− ωex1 + �αr(Mz2 − x2)]] (54)

aking into account that all states and parameters of inductionotor are bounded, then, it exists sufficiently large positive num-

W

ms Research 77 (2007) 604–618

ers η1 and η2 (satisfying condition of Eq. (52)) such that

T(

dS

dτ

)< 0 �

emark (3). Once the currents trajectory reaches the slidingurface S = ez = 0, the error system of (46), in sliding mode,ehaves as a reduced-order subsystem governed only by theotor flux error ex assuming that ez = 0 and ez = 0.

.5. Slow reduced-order error dynamics (rotor fluxbserver)

For slow error dynamics (when S ≡ 0, i.e., ez = ez = 0), wese the system (46) and setting ε = 0. So, we can write

= μ[(αrI2 − ωJ2)ex + �αr(Mz − x)] − GzIs (55)

˙x = −[(αrI2 − ωJ2)ex + �αr(Mz − x)] − GxIs (56)

.6. Stability analysis of the slow reduced-order errorynamics

roposition (3). Consider the slow reduced-order errorynamics given by (55) and (56), and the current observer gainatrix Gz of (50). Assuming that the inverse of the rotor time-

onstant is an unknown. So, if the flux observer gain matrix Gx

s chosen as

x =((

q2αr − 1

μ

)I2 + q2ωJ2

)Gz (57)

with the adaptive law

d

dt�αr = −q1q2(GzIs)

T(Mz − x) (58)

here q1 > 0 and q2 > 0, then the flux observer is globally stable.

roof (3). System of (55) and (56) becomes

= μ[Aex + �αrB] − H (59)

˙x = −[Aex + �αrB] − LH (60)

here

H = GzIs, L = GxG−1z ,

A = (αrI2 − ωJ2), B = (Mz − x).

Let we choose the candidate Lyapunov function as follows

= 1

2(ex)Tex + 1

q1

1

2(�αr)

2 (61)

here q1 > 0.

The t-time-derivative of W can be expressed as

˙ = (ex)Tex + �αr

1

q1

d

dt�αr (62)

Systems Research 77 (2007) 604–618 611

F

e

i

e

o

e

�

W

w

Q

n

−a

�

Table 1Nominal parameters of induction motor

1.5 kW 220/380 V 3.68/6.31 A

N = 1420 rpm Rs = 4.85 � Rr = 3.805 �

Lp

Q

gc

o

6

cws

A. Mezouar et al. / Electric Power

We know that W is positive definite. Now, look for ex and ex.rom Eq. (59), we get:

x = A−1(

1

μH − �αrB

)(63)

and with addition, member to member, of Eqs. (59) and (60),t yields:

˙x = − 1

μ(I2 + μL)H (64)

r

˙x = −ΛH (65)

where

= 1

μ(I2 + μL) (66)

Now, substituting (63) and (65) in (62), it yields:

˙ = − 1

μHTQH + �αrH

TQB + �αr1

q

d

dt�αr (67)

here

= �TA−1 (68)

Since μ > 0, sufficient condition for (67) to be negative defi-ite will be satisfied if

HTQH < 0 (69)

nd

αr

(HTQB + 1

q1

d

dt�αr

)= 0 (70)

ooDt

Fig. 1. Block diagram of the propos

s = 0.274 H Lr = 0.274 H M = 0.258 H= 2 J = 0.031 kg m2 f = 0.00113 N m s/rad

The condition of (69) is verified choosing

= ΛTA−1 = q2I2 (71)

where q2 > 0 and the substitution of (71) in (70) permits toive a simple adaptive law for the inverse of the rotor time-onstant estimation as follows

d

dt�αr = −q1q2H

TB (72)

r

d

dt�αr = −q1q2(GzIs)

T(Mz − x)

as given in Eq. (58). �

. Simulation results

The proposed algorithm for the rotor flux and rotor time-onstant estimation has been simulated for the induction motorhose data are given in Table 1. The block diagram of the control

ystem is shown in Fig. 1. As a controller, the indirect-field-

riented sliding mode control is used. Sliding mode control andbserver parameters are listed in Tables 2 and 3, respectively.ue to the high harmonic content of the voltages provided by

he inverter, the observer input voltages are the reference v∗sα

ed control-observer structure.

612 A. Mezouar et al. / Electric Power Systems Research 77 (2007) 604–618

Fig. 2. Sensitivity of the system performance to changes on the rotor resistance first by 50% and next by 100% with φ∗rd = 1.0 Wb: (a) reference signals of rotor

resistance (solid) and load torque (dotted), (b) reference (dotted) and estimated (solid) rotor resistance, (c) reference (dotted) and actual (solid) speeds, (d) zoom onreference (dotted) and actual (solid) speeds, (e) rotor fluxes estimation: φrd (solid) and φrq (dotted), (f) rotor fluxes error: eφrd (solid) and eφrq (dotted), (g) estimatedload torque. (h) Stator currents error: eisα (solid) and eisβ (dotted).

A. Mezouar et al. / Electric Power Systems Research 77 (2007) 604–618 613

Fig. 3. Sensitivity of the system performance to change in the external load with Rr = 1.25 Rrn and φ∗rd = 1.0 Wb: (a) real (dotted) and estimated (solid) load torques,

(b) reference (dotted) and actual (solid) speed, (c) reference (dotted) and estimated (solid) rotor resistance, (d) zoom on reference (dotted) and estimated (solid) rotorresistance, (e) load torque (dotted) and motor torque (solid), (f) stator currents error: eisα (solid) and eisβ (dotted), (g) rotor fluxes estimation: φrd (solid) and φrq

(dotted), (h) rotor fluxes error: eφrd (solid) and eφrd (dotted).

614 A. Mezouar et al. / Electric Power Syste

Table 2Sliding mode control parameters

λω 120λφ 120Kω 80Kφ 80δω = δ1 0.5δω = δ2 0.5

Table 3Sliding mode observer parameters

η1 50η2 50q1 5000q

ad

mrtrua

ap

bh

6

itmT1mtFgtagmTposed observer can track the reference command accurately and

F(

2 0.002

nd v∗sβ. For the slip frequency estimation (SFE) of Eq. (25), the

-axis rotor flux is replaced with the rated rotor flux value φ∗r .

It is considered that the load torque is unknown and all theotor parameters are known and constant except for the rotor

esistance which will change during the motor operating. Forhis closed loop system, rotor flux feedback signal and rotor

esistance are replaced with the estimate corresponding val-es provided by the adaptive sliding mode observer. With thessumption that all states including rotor flux and all parameters

qir

ig. 4. Sensitivity of the system performance to change in the reference speed with Rb) reference (dotted) and estimated (solid) rotor resistance, (c) estimated rotor fluxes:

ms Research 77 (2007) 604–618

re available, rotor flux and rotor resistance estimated by theroposed method are compared to their actual values.

In order to verify the performance of the field-oriented controlased on the proposed sliding mode observer, the control systemas been tested in several operating conditions.

The results are summarized in this section:

.1. Rotor resistance variation effect

This test consists in increasing the rotor resistance. As shownn Fig. 2(a), the motor is started with its nominal rotor resis-ance value Rrn = 3.805 �. Then, the rotor resistance of the motor

odel is suddenly set to 1.5 Rrn at t = 1 s, and to 2Rrn at t = 2 s.he reference speed and reference rotor-flux are maintained at400 rpm and 1.0 Wb, respectively. Fig. 2(b) compares the esti-ated and actual rotor resistance. After a short convergence

ime, the estimated rotor resistance reaches the actual value.ig. 2(c) and (d) show the speed response of the motor; a veryood speed regulation is obtained. In Fig. 2(e) and (f), are shownhe estimated rotor fluxes and the error between them and thectual values. It can be noticed the high flux tracking and theood rotor flux orientation. Fig. 2(g) and (h) show the esti-ated load torque and the stator currents error, respectively.hese results show that the sliding mode control with the pro-

uickly. It is important to notice that the q-axis rotor flux errors greater than the d-axis rotor flux error in transient state. Thiseport is very clear since the rotor resistance estimation error (in

r = 1.25 Rrn and φ∗rd = 1.0 Wb: (a) reference (dotted) and actual (solid) speed,

φrd (solid) and φrq (dotted), (d) rotor fluxes error: eφrd(solid) and eφrq (dotted).

Syste

ta

6

iawpE

ltspagm

F((e

A. Mezouar et al. / Electric Power

ransient state) propagate on the slip frequency which directlyffects the rotor field orientation (see Eq. (25)).

.2. Performance under external load disturbances

The observer sensitivity to external load disturbances is alsonvestigated in this study. The objective is to follows: the speed

nd rotor flux references in spite of disturbances in load torqueith a constant error (of +25%) in the rotor resistance value. Thisractical error is made to test the efficacy of the adaptive law ofq. (58). Fig. 3(a) shows the actual and the estimated applied

rlpo

ig. 5. Sensitivity of the system performance to changes on the rotor resistance first ba) reference signals of rotor resistance (solid) and load torque (dotted), (b) referencesolid) speed, (d) stator currents error: eisα (solid) and eisβ (dotted), (e) rotor fluxes

φrd (dotted).

ms Research 77 (2007) 604–618 615

oad torque. Due to the transient rotor resistance estimation error,he estimated load torque presents negative values in the motortart-up and after it follows exactly the actual signal. Fig. 3(b)resents very good performance for speed regulation. Fig. 3(c)nd (d) present the actual and estimated rotor resistance. A veryood estimation is obtained. In Fig. 3(e) and (f) are shown theotor and the real load torques, and the stator currents error,

espectively. The current observer is insensitive to this externaload variation. Fig. 3(g) and (h) show that the completely decou-led control of rotor flux and torque is obtained and the globalbserver is very robust to external load disturbances.

y 50% and next by 100% with an error of +50% on the motor stator resistance:(dotted) and estimated (solid) rotor resistance, (c) reference (dotted) and actualestimation: φrd (solid) and φrq (dotted), (f) rotor fluxes error: eφrd (solid) and

6 Syste

6

w

aTFirtara

tw

6

trao

F((

16 A. Mezouar et al. / Electric Power

.3. Performance over wide speed range

In this case, we consider the speed tracking performance foride variation range of reference speed.The rotor flux reference is kept at its rated value of 1.0 Wb

nd the motor is operating without external load disturbances.he observer performance for speed tracking is presented inig. 4(a). The actual and the estimated rotor resistance are shown

n Fig. 4(b). At very low speed, the rotor resistance error isemarkable, but it remains in the accepted values range. On

he other hand, the rotor resistance estimation is very goodt high speed. Fig. 4(d) and (c) show the estimation of theotor fluxes and the error between the estimated rotor fluxesnd the actual rotor fluxes, respectively. These results prove that

rerr

ig. 6. Sensitivity of the system performance to changes on the rotor resistance first bsolid) and load torque (dotted), (b) reference (dotted) and estimated (solid) rotor resise) rotor fluxes estimation: φrd (solid) and φrq (dotted), (f) rotor fluxes error: eφrd (sol

ms Research 77 (2007) 604–618

he speed tracking is quite good and the rotor-field is alwaysell-oriented.

.4. Stator resistance uncertainty effect

The following test consists of applying an unmodeled value ofhe motor stator resistance of 1.5 Rsn where the controller statoresistance is always set to Rsn, and again examine the same tests in Fig. 2. As shown in Fig. 5(b) and (f), an incorrect valuef the stator resistance will influence the adaptation of the rotor

esistance and the rotor flux estimation. So, the rotor resistancestimation presents a constant error due to the unmodeled statoresistance, but this error is less than 2.5%. Also, the estimatedotor fluxes present a constant error due to the incorrect rotor

y 50% and next by 100% with J = 2Jn: (a) reference signals of rotor resistancetance, (c) reference (dotted) and actual (solid) speeds, (d) estimated load torque,id) and eφrq (dotted).

Syste

tar

6

uFFcctetvb

7

mfartsspgvwt

A

Lfi

iIIJLMRSSSSSTTU

v

v

x

x

x

Gα

φ

φ

ω

ω

R

[

[

[

[

[

[

[

[

[

A. Mezouar et al. / Electric Power

ime-constant estimation and the decoupling control is partiallyffected. As expected, the increase in the stator resistance givesise to the rotor resistance estimation.

.5. Rotor inertia uncertainty effect

The last test investigates the sensitivity of the observer againstncertainties in the mechanical parameters. The same test ofig. 2 is reproduced assuming uncertainty in the rotor inertia (J).ig. 6 illustrates the responses of the proposed test for a coeffi-ient J = 100% higher than its nominal value. From Fig. 6(b), itan be seen that the rotor resistance estimation is insensitive tohis uncertainty. Fig. 6(e) and (f) show that the rotor flux is well-stimated and completely oriented. As seen, the uncertainty ofhe parameter J does not affect the steady-state of the estimatedalues. It modifies only the transient behavior of the observer,ut it does not compromise its stability.

. Conclusions

Based on the singular perturbation theory, a robust slidingode rotor flux observer with rotor time-constant estimation

or induction motor has been presented. The rotor flux observerccuracy is guaranteed through the stator currents observer. Theotor time-constant is estimated using the stator currents estima-ion error and the observed rotor flux, based on the Lyapunovtability theory. The efficiency of the control-observer structurecheme has been successfully verified by simulation. The pro-osed sliding mode observer-based control demonstrated veryood performance, especially; it is robust under rotor resistanceariation, external load disturbances and speed tracking. Futureork is oriented at experimental validation, including stator

ime-constant estimation.

ppendix A

In this paper, the following notations are used:

ist of symbolscoefficient of viscous friction

sd, isq stator currents in the synchronous rotating referenceframe

sα, isβ stator currents in the stationary reference frame.M induction motors switching vector, p moment of rotor inertia, numbers of pole pairss, Lr stator and rotor inductances, σ mutual inductance and leakage factor

s, Rr stator and rotor resistances.M.C sliding mode control.M.O sliding mode observer.F.E slip frequency estimationc control sliding surface

observer sliding surfaces, Tr stator and rotor time-constantse, TL electromagnetic and load torques, V, W Lyapunov functions

[

ms Research 77 (2007) 604–618 617

sd, vsq stator voltages in the synchronous rotating referenceframe

sα, vsβ stator voltages in the stationary reference frameˆ, x* estimated and reference value of x˙ t time-derivative of x (dx/dt)n nominal value of x

reek lettersr inverse of the rotor time-constant αr = Rr/Lr

rd, φrq rotor fluxes in the synchronous rotating reference framerα, φrβ rotor fluxes in the stationary reference frame, ω* electrical rotor speed and reference rotor speeds, ωsl synchronous frequency, slip frequency ωsl = ωs − ω

eferences

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

[2] P. Vas, Vector Control of AC Machines, Oxford University Press, London,UK, 1990.

[3] V.I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Aut.Cont. AC-22 (1977) 212–222.

[4] A. Sabanovic, F. Bilalovic, Sliding-mode control of ac drives, IEEE Trans.Ind. Appl. IA-25 (1) (1989) 70–75.

[5] V.I. Utkin, Sliding mode control design principles and applications to elec-tric drives, IEEE Trans. Ind. Electron. 40 (1993) 23–36.

[6] C.C. Chan, H.Q. Wang, New scheme of sliding mode control for highperformance induction motor drives, IEE Proc. Electr. Power Appl. 143(3) (1996) 177–185.

[7] L.G. Shiau, J.L. Lin, Stability of sliding mode current control for highperformance induction motor position drives, IEE Proc. Electr. Power Appl.148 (1) (2001) 69–75.

[8] A. Benchaib, A. Rachid, E. Audrezet, M. Tadjine, Real-time sliding-modeobserver and control of an induction motor, IEEE Trans. Ind. Electron. 46(1) (1999) 128–138.

[9] G.C. Verghese, S.R. Sanders, Observers for flux estimation in inductionmachines, IEEE Trans. Ind. Electron. 35 (1) (1988) 85–94.

10] J.A. Santisteban, R.M. Stephan, Vector control methods for inductionmachines: an overview, IEEE Trans. Educ. 44 (2) (2001) 170–175.

11] H.A. Toliyat, E. Levi, M. Raina, A review of RFO induction motor param-eter estimation techniques, IEEE Trans. Energy Conversion 18 (2003)271–283.

12] P.L. Jansen, R.D. Lorenz, D.W. Novotny, Observer-based direct field orien-tation: Analysis and comparison of alternative methods, IEEE Trans. Ind.Appl. 30 (1994) 945–953.

13] M. Hinkkanen, Analysis and design of full-order flux observers for sensor-less induction motors, IEEE Trans. Ind. Electron. 51 (5) (2004) 1033–1040.

14] T.O. Kowalska, Application of extended Luenberger observer for flux androtor time-constant estimation in induction motor drives, IEE Proc. 136 (6)(1989) 324–330.

15] T. Du, P. Vas, F. Stronach, Design and application of extended observersfor joint state and parameter estimation in high-performance AC drives,IEE Proc. Electr. Power Appl. 142 (2) (1995) 71–78.

16] L.C. Zai, C.L. Demarco, T.A. Lipo, An extended Kalman filter approachto rotor time-constant measurement in PWM induction motor drives, IEEETrans. Ind. Appl. 28 (1) (1992) 96–104.

17] D.J. Atkinson, J.W. Finch, P.P. Acamley, Estimation of rotor resistance ininduction motor, IEE Proc. Electr. Power Appl. 143 (1) (1996) 87–94.

18] K. Ohnishi, Y. Ueda, K. Miyachi, Model reference adaptive system against

rotor resistance variation in induction motor drive, IEEE Trans. Ind. Elec-tron. IE-33 (1986) 217–223.

19] H. Sugimoto, S. Tamai, Secondary resistance identification of an inductionmotor applied model reference adaptive system and its characteristics, IEEETrans. Ind. Appl. IA-23 (1987) 296–303.

6 Syste

[

[

[

[

[

[

[

[

[

[

[

18 A. Mezouar et al. / Electric Power

20] Y. Koubaa1, M. Boussak, Rotor resistance tuning for indirect stator flux ori-entated induction motor drive based on MRAS scheme, Eur. Trans. Electr.Power 15 (2005) 557–570.

21] H.J. Shieh, K.K. Shyu, F.J. Lin, Adaptive estimation of rotor time-constantfor indirect field-oriented induction motor drive, IEE Proc. Electr. PowerAppl. 145 (1998) 111–118.

22] S.H. Jeon, K.K. Oh, J.Y. Choi, Flux observer with online tuning of statorand rotor resistances for induction motors, IEEE Trans. Ind. Electron. 49(2002) 653–664.

23] C. Aurora, A. Ferrara, A. Levant, Speed regulation of induction motors:a sliding mode observer-differentiator based control scheme, in: Proceed-ings of the 40th IEEE Conference Dec. Contr, Florida, USA, 2001, pp.2651–2656.

24] N. Inanc, A new sliding mode flux and current observer for direct field ori-ented induction motor drives, Electr. Power Syst. Res. 63 (2002) 113–118.

25] B. Karanayil, M. Fazlur Rahman, C. Grantham, Stator and rotor resistanceobservers for induction motor drive using fuzzy logic and artificial neuralnetworks, IEEE Trans. Energy Conversion 20 (4) (2005) 771–780.

[

ms Research 77 (2007) 604–618

26] P.V. Kokotovic, H. Khalil, J. O‘Reilly, Singular Perturbation Methods inControl: Analysis and Design, Academic Press, New York, 1986.

27] P.V. Kokotovic, J. O‘Mailley, P. sannuti, Singular perturbation and orderreduction in control theory: an overview, Automatica 12 (1976) 123–132.

28] H. Hofmann, S.R. Sanders, Speed sensorless vector control of inductionmachines using two-time-scale approach, IEEE Trans. Ind. Appl. 34 (1)(1998) 169–177.

29] M. Djemai, J. Hernandez, J.P. Barbot, Non-linear control with fluxobserver for a singularly perturbed induction motor, in: Proceedingsof the 32th IEEE Conference Dec. Contr, USA, 1993, pp. 3391–3396.

30] J. De-leon, Jm. Alvares, R. Castro, Sliding mode control and state estima-tion for non-linear singularly perturbed systems: application to an induction

electric machine, in: Proceedings of the 34th IEEE Conference Control.App, New York, 1995, pp. 998–1003.

31] J. De-leon, R. Castro, Jm. Alvares, Two-time sliding mode control and stateestimation for non-linear systems, in: Proceedings of the 13th TriennialWorld Cong, IFAC, San Francisco, USA, 1996, pp. 265–270.