Robust sliding mode control and flux obs

Keywords Induction motor drives· Singular perturbation theory· Sliding mode control · Sliding mode observer List of Symbols ω, ω∗ Electrical rotor and reference speeds v sd , v sq Stator

O R I G I N A L PA P E R

A Mezouar · M K Fellah · S Hadjeri

Robust sliding mode control and flux observer for induction motor using singular perturbation

Received: 18 July 2005 / Accepted: 20 September 2005 / Published online: 24 February 2006

© Springer-Verlag 2006

Abstract This paper proposes a sequential methodology for

designing a robust adaptive sliding mode observer for an

induction motor drive using a two-time-scale approach This

approach is based on the singular perturbation theory The

two-time-scale decomposition of the original system of the

observer error dynamics into separate slow and fast

subsys-tems permits a simple design and sequential determination

of the observer gains In the proposed method, the stator

cur-rents and rotor flux are observed on the stationary reference

frame using the sliding mode concept The control algorithm

is based on the indirect field oriented sliding mode control

with an on-line adaptation of the rotor resistance to keep the

machine field oriented The control–observer scheme seeks

to provide an asymptotic tracking of speed and rotor flux

in spite of the presence of an uncertain load torque and an

unknown value of rotor resistance The validity for practical

implementation has been verified through computer

simula-tions

Keywords Induction motor drives· Singular perturbation

theory· Sliding mode control · Sliding mode observer

List of Symbols

ω, ω∗ Electrical rotor and reference speeds

v sd , v sq Stator voltages in the synchronously rotating

reference frame

i sd , i sq Stator currents in the synchronously

rotating reference frame

φ rd , φ rq Rotor fluxes in the synchronously rotating

reference frame

A Mezouar (B) · M K Fellah · S Hadjeri

Intelligent Control and Electrical Power Systems Laboratory,

Department of Electrical Engineering,

Faculty of Sciences Engineering,

Djillali Liabes University,

22 000 Sidi Bel Abb`es, Algeria

Tel.: +213-72-320609

Fax: +213-48-474262

E-mail: a mezouar@yahoo.com

E-mail: mkfellah@yahoo.fr

E-mail: shadjeri@yahoo.fr

v sα , v sβ Stator voltages in the stationary

reference frame

i sα , i sβ Stator currents in the

stationary reference frame

φ rα , φ rβ Rotor fluxes in the

stationary reference frame

ωs, ωsl Synchronous frequency,

slip frequency ωsl = ωs− ω

Ls, Lr Stator and rotor inductances

Rs, Rr Stator and rotor resistances

Ts, Tr Stator and rotor time-constants

αr Rotor inverse time-constant αr = Rr/Lr

M, σ Mutual inductance and leakage factor

J, p Moment of inertia of the rotor and numbers

of pole pairs

f Coefficient of viscous friction

Te, TL Electromagnetic and load torques

Sc Control sliding surface

S Observer sliding surface

∧

(.), (.)∗ Estimated and reference value of (.)

•

(.) t time-derivative of (.) (d(.)/dt)

1 Introduction

The control of the induction motor has attracted much atten-tion in the past few decades One of the most significant developments in this area was the field oriented control The orientation of the flux made it possible to act independently

on the rotor flux and the electromagnetic torque through the intermediary of the components of the stator voltage [1, 2] Unfortunately, this control approach suffers from its sensitiv-ity to the motor parameter variations When the motor param-eters change with temperature and magnetic saturation, the performance of the system will deteriorate There are two ways to solve this problem The first is to perform an on-line identification of the motor parameters and accordingly update the values used in the controller The other solution

is to use a robust control algorithm (i.e., insensitive to the

motor parameter variations)

In the last years, the sliding mode technique has been

widely studied and developed for the control and state

esti-mation problems since the studies of Utkin [3] This

con-trol technique allows a good steady state and good dynamic

behavior in the presence of system parameters’ variation and

disturbances [4–6] Several methods of applying the sliding

mode control to induction motor drives have been presented

[4–7] All of these methods have a common feature: the

anal-ysis and design of the sliding mode controller are based on

the mathematical model of the induction motor as used in

indirect vector control

Most control schemes for the induction motor require

accurate information on state variables and parameters

There-fore, fluxes are usually estimated with measured stator

cur-rents, stator voltages, and motor speed by using the current

model for rotor flux or voltage model for stator flux [8, 9]

However, since the estimated fluxes are basically dependent

on the motor model, a variation in the parameters inevitably

propagates to the flux estimated error

Among these parameters, rotor resistance varies mainly

depending on the temperature One of the solutions to reduce

the estimation error is to design an adaptive observer

compen-sating for the variation To overcome this problem, numerous

on-line identification schemes dealing with rotor resistance

have been proposed Some researches have proposed

var-ious induction motor drives with rotor resistance or rotor

time-constant identification [10–16] to produce better

con-trol performance, such as a model reference adaptive system,

Luenberger observer, and the extended kalman filter

On the other hand, the singular perturbation theory

pro-vides the mean to decompose two-time-scale systems into

slow and fast subsystems of lower order described in separate

time-scales, which greatly simplify their structural analysis

and any subsequent control design Then, the control (and/or

observer) design may be carried out for each lower order

sub-system, and the combined results yield a composite control

(and/or observer) for the original system [17–19]

So, the idea of combining the singular perturbation theory

and the sliding mode technique constitutes a good possibility

to achieve classical control objectives for systems having

un-modeled or parasitic dynamics and parametric uncertainties

[20–22]

In this paper, an adaptive sliding mode observer is

devel-oped for the simultaneous estimation of the rotor flux

com-ponents and of the rotor resistance for an induction motor

under the assumption that only the stator currents and the

motor speed are available for measurement Singular

per-turbation theory is used for having a sequential and simple

design of this observer In addition, an adaptive law based on

the Lyapunov stability theory estimates the rotor resistance

This paper is organized as follows: the main results of the

time-scales approach and the design of the general

two-time-scales sliding mode observer are presented in Sects 2

and 3, respectively In Sect 4, we briefly review the indirect

field oriented sliding mode control of the induction motors

The design of a two-time-scale sliding mode observer for the induction motor model is presented in Sect 5 In that section, a study of the stability analysis of this observer is made via the singular perturbation method with the sliding mode concept and the Lyapunov stability theory In Sect 6, and through simulation, the studied observer is associated to the indirect field oriented sliding mode control where rotor fluxes and rotor resistance are replaced by those delivered by the observer Finally, in Sect 7, we provide some comments and conclusions

2 Two-time-scale approach

The two-time-scale approach, based on the singular pertur-bation theory, can be applied to systems where the state vari-ables can be split into two sets, one having “fast” dynamics, and the other having “slow” dynamics The difference be-tween the two sets of dynamics can be distinguished by the

use of a small multiplying scalar ε Generally, the scaling parameter ε is the speed ratio of the slow versus fast

phe-nomena

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

where t0is the initial time

The reader is referred to [17] and [18] for the general theory on singular perturbation

2.1 Nonlinear singularly perturbed systems Let us consider the following class of nonlinear singularly perturbed systems described by the so-called standard singu-larly perturbed form:

d

dt x = fx(x, z, u, t, ε), x(t0) = x0,

ε d

dt z = fz(x, z, u, t, ε), z(t0) = z0, (2) where x ∈  n is the slow state, z ∈  m is the fast state,

u ∈  p is the control input and ε is a small positive parame-ter such that ε ∈ [0, 1] f x and f zare assumed to be bounded and analytic real vector fields, and we consider a vector of measurement that is linearly related to the fast state vector as

2.2 Slow reduced subsystem

In the limiting case, as ε → 0 in (2), the asymptotically stable

fast transient decays ‘instantaneously’, leaving the

reduced-order model in the t time-scale defined by the quasi-steady states x s (t) and zs (t):

d

and the substitution of a root of (5)

zs = h(x s , us, t), (6)

into (4) yields a reduced model

d

dt xs = f x (xs , h(xs , us, t), us , t, 0), xs (t0) = x0, (7)

where the index (s) indicates that the associated quantity

belongs to the system without ε.

2.3 Fast reduced subsystem

The fast dynamic subsystem (also known as the boundary

layer system) denoted by z f, which represents the derivation

of z s from z, is obtained by transforming the slow time-scale

t to the fast time-scale τ = (t − t0)/ε System of Eq (2)

becomes

d

dτ x = ε fx (x, z, u, ετ + t0),

d

Introducing the derivation of z s from z, i.e., z f = z − z s,

and again examining the limit as ε → 0, it yields

d

dτ zf = f z

x0, zs(0) + zf (τ ), uf (τ ), t0



with

zf (0) = z0− z s(0),

where u f = u − u s is the fast part of the input control

2.4 Two-time-scale states approximation

Fast and slow variables given by (7) and (9) can be

com-bined into a composite structure in order to approximate the

original states of (2) as given in [17]:

x = xs(t) + O(ε),

3 Two-time-scale sliding mode observer

Consider the above continuous nonlinear singularly perturbed

system of Eq (2) which is given by



˙x = f x (x, z, u, ε)

In addition, it is assumed that the above system is

observ-able Consequently, the observer design may be considered

for the state observation of slow variables from the

measure-ment of fast variables [20–22]

3.1 Sliding mode observer design

By structure, an observer based on the sliding mode approach

is very similar to the standard full order observer with replace-ment of the linear corrective terms by a discontinuous func-tion [5–7]

The corresponding sliding mode observer for the system

of (11) can be written as a replica of the system with an additional nonlinear auxiliary input term as follows:



˙ˆx = f x ( ˆx, z, u, ε) + Gx s

where  s = sign
S(y, ˆy)

is the switching function, and G x and G z are the observer gains with (n × m) and (m × m)

dimensions, respectively, to be determined

The sliding surface function S can be chosen as a linear function of (y − ˆy) as given in [7], so

where (y − ˆy)T=(y1− ˆy1) (y2− ˆy2) · · · (ym − ˆy m)

and

is a (n × m) gain matrix to be specified.

The error dynamics is calculated by subtracting (12) from (11):



˙e x = f x (x, z, u, ε) − fx ( ˆx, z, u, ε) − Gx s

ε ˙ez = f z(x, z, u, ε) − fz( ˆx, z, u, ε) − Gzs , (14)

or



where

ex = x − ˆx,

ez = z − ˆz,

x = f x (x, z, u, ε) − fx ( ˆx, z, u, ε),

z = f z(x, z, u, ε) − fz( ˆx, z, u, ε).

The observer gains design can be based on the sequential application of the resulted subsystems of (15) by applying the singular perturbation methodology We first need to analyze the fast variables tracking properly using the so-called reach-ing condition (based on measurable state variables), and, thereafter, the slow variables’ asymptotic-convergence (for inaccessible state variables)

3.2 Stability analysis in the fast time-scale For the fast error dynamic subsystem, the associated

time-scale is defined by τ = (t − t0 )/ε; then (15) can be

trans-formed into







de x

dτ x − G x s )

de z

dτ z − G zs.

(16)

Setting ε = 0 in (16), it yields

de z

with

de x

dτ = 0.

In this time-scale, the stability analysis involves the

deter-mination of the observer gain G z so that in this time-scale

(τ ), the surface S(τ ) = 0 is attractive.

It can be shown that when the sliding mode occurs on

S(τ ), the equivalent value of the discontinuous observer

aux-iliary input is found by solving Eq (17) for G zsafter

insur-ing a value of zero for de z dτ such as

Gz ˜ s z,

and the equivalent switching vector is obtained as

˜ s = G−1

3.3 Stability analysis in the slow time-scale

The slow error dynamic subsystem can be found by

consid-ering ε = 0 in (15); so

de x

From (20), the equivalent switching vector can be

re-found as

˜ s = G−1

z z.

Therefore, by an appropriate choice of G x, the desired

rate of convergence e x → 0 can be obtained

4 Sliding mode control review of the induction motor

Assuming that the induction motor model system is

control-lable and observable, the sliding mode control consists of two

phases [4, 5]:

• designing an equilibrium surface, called the sliding surface,

such that any state trajectory of the plant restricted to the

sliding surface is characterized by the desired behavior;

and

• designing a discontinuous control law to force the system

to move on the sliding surface in a finite time

4.1 Dynamic model of induction machine

Under the assumptions of linearity of the magnetic circuit

and neglecting iron losses, the state space model of the

three-phase induction motor expressed in the synchronously

rotat-ing reference frame (d − q) is























d

dt i sd= −R λ

σ Ls

i sd + ωsi sq+ µ

σ Ls

1

Tr

σ Ls

σ Ls

v sd

d

dt i sq = −ωsi sd− R λ

σ Ls

i sq− µ

σ Ls

σ Ls

1

Tr

σ Ls

v sq

d

dt φ rd= M

Tri sd− 1

Trφ rd + ωslφ rq

d

dt φ rq= M

Tr

i sq − ωslφ rd− 1

Tr

φ rq

dω

dt =p

J (Te− T L ) − f

J ω,

(21)

with the constants defined as follows:

Rλ = Rs+M2

L2 r

Rr, σ = 1 − M

2

LsLr,

Lr, where the state variables are the stator currents (isd , i sq ), the rotor fluxes (φrd , φ rq ) and the rotor speed ω, and the stator voltages (vsd , v sq ) and slip frequency ωslare the control vari-ables The electromagnetic torque expressed in terms of the state variables is

Te= pM

4.2 Rotor field oriented induction motor model Among the various sliding mode control solutions for the induction motor proposed in the literature, the one based on indirect field orientation can be regarded as the simplest one Its purpose is to directly control the inverter switching by the use of two switching surfaces

The induction motor equations in the synchronously

rotat-ing reference frame (d − q), oriented in such a way that the rotor flux vector points into d-axis direction, are the

follow-ing:



















d

dt ω = f1

d

dt φ rd = f2 d

dt i sd = f3+ 1

σ Lsv sd

d

dt i sq = f4+ 1

σ Lsv sq

(23)

with

ωsl= M

Tr

i sq

where













f1= k cφ rd i sq −p

J TL−f

J ω

f2= M

Tri sd − 1

Trφ rd

f3= −R λ

σ Lsi sd + ωsi sq+ µ

σ Ls

1

Trφ rd

f4= −ωsi sd− R λ

σ L i sq− µ

σ L ωφ rd ,

(25)

kc= p2M

J Lr.

4.3 Speed and flux sliding mode controller

Using the reduced nonlinear induction motor model of

Eq (23), it is possible to design both a speed and a flux

sliding mode controller Let us define the sliding surfaces







Sc1= Sc1(ω) = λω(ω∗− ω) + d

dt (ω

∗− ω)

Sc2= Sc2(φr) = λφ(φ∗r − φ dr) + d

dt (φ

∗

r − φ dr),

(26)

where λ ω > 0, λφ > 0, ω∗and φr∗are the reference speed

and the reference rotor flux, respectively

To determine the control law that leads the sliding

func-tions (26) to zero in finite time, one has to consider the

dynam-ics of Sc = (Sc1, Sc2)T, described by

where

F =



 ¨ω∗+λ ω ˙ω∗+

f

J ˙TL

 + −λ ω+f

J



f1−k c (i sq f2+ φ rd f4) ( ¨ φ∗r + λ φ ˙φ∗

r) + −λ φ+ 1

Tr



f2 −M

Trf3



,

σ Ls



k c φ rd 0

0 M/Tr



, V S=



v sq

v sd



.

If the Lyapunov theory of stability is used to ensure that

Scis attractive and invariant, the following condition has to

be satisfied

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

stator voltages as follows:



v sq

v sd



 = −D−1F − D−1



0 K φ







sign(Sc1 ) sign(Sc2 )



, (29) where

Proof 1 see Appendix

The sliding mode causes drastic changes in the control

variables introducing high frequency disturbances To reduce

the chattering phenomenon, a saturation function sat(Sc )

in-stead of the switching one sign(Sc ) has been introduced

sat(Sc i ) =

S c i

δ i if|(S c i )| ≤ δi

sign(Sc i ) if|(S c i )| > δi ,

(31)

Remark 1

• From the above control law of Eq (29), it can be seen

that the implementation of these algorithms requires load torque and rotor flux estimations since stator currents, sta-tor voltages and rosta-tor speed are available by measures

In the next section, we focus on by a robust estimation

of rotor flux The estimated load torque can be easily obtained by using the mechanical equation of the motor model with estimated rotor fluxes and measured stator currents

• In the following, we assume to operate with constant

ref-erence speed, constant refref-erence rotor flux and constant load torque, so that ˙ω∗= 0, ˙φ∗

r = 0 and ˙TL= 0.

5 Two-time-scale sliding mode observer design for the induction motor

Consider only the first four equations of the induction motor model of Eq (21) in which the speed motor will be con-sidered as a time-varying parameter The objective of the studied observer is to estimate the unmeasured rotor fluxes The sliding mode observer design procedure comprises of the following two steps [7]:

• designing an equilibrium surface such that the estimation

error trajectories restricted to this surface have the desired stable dynamics; and

• determining the observer gains to drive the estimation

error trajectories to the sliding surface and maintain it on the set

5.1 Dynamic model of the induction motor Using the model of Eq (21), the state space model of the induction motor, without mechanical equation, expressed in

the fixed stator reference frame (α, β) is



















d

dt i sα = − Rλ

σ Lsi sα+ µ

σ Ls

1

Trφ rα+ µ

σ Lsωφ rβ+ 1

σ Lsv sα

d

dt i sβ = − Rλ

σ Ls

i sβ− µ

σ Ls

ωφ rα+ µ

σ Ls

1

Tr

φ rβ+ 1

σ Ls

v sβ

d

dt φ rα = M

Tri sα− 1

Trφ rα − ωφ rβ

d

dt φ rβ = M

Tri sβ + ωφ rα− 1

Trφ rβ

(32)

Voltage, current and flux transformation from the syn-chronous to the stationary reference frame and vice versa is made as [1, 2]:



xα xβ



=



cos(θ s ) − sin(θs) sin(θ s ) cos(θs)

 

xd xq



and



xd xq



=



cos(θ s) sin(θ s )

− sin(θ s ) cos(θs )

 

xα xβ



where x = v, i, φ, and θ sis the angular displacement of the

synchronously rotating reference frame

5.2 Singularly perturbed induction motor model

Based on the well-known of the induction machine model

dynamics [20, 21], the slow variables are (φrα , φ rβ ) and the

fast variables are (isα , i sβ ) Therefore, the corresponding

stan-dard singularly perturbed form with ε = σ Ls, x = (φrα , φ rβ )T

and z = (isα , i sβ )Tis











ε˙z1= −R λz1+ µαrx1+ µωx2+ v sα

ε˙z2= −R λz2− µωx1+ µαrx2+ v sβ

˙x1 = Mαrz1− αrx1− ωx2

˙x2 = Mαrz2+ ωx1− αrx2,

(35)

where

αr= 1

Tr = Rr

Lr, Rλ = Rs+ M µ αr.

Remark 2

• All parameters of the induction motor will be considered

as constant except for the rotor resistance The rotor

resis-tance Rr will be treated as an uncertain parameter with Rrn

as its nominal value An additional assumption is that Rr

varies slowly (practical assumption), so that ˙Rr≈ 0

• The motor speed will be treated as a bounded time-varying

variable

5.3 Singularly perturbed sliding mode observer

From Sect 3, the observer equations of the above model

based on the sliding mode concept are the following















ε ˙ˆz1= − ˆR λz1+ µ ˆαrˆx1+ µω ˆx2+ v sα + G z1s

ε ˙ˆz2= − ˆR λz2− µω ˆx1+ µ ˆαrˆx2+ v sβ + G z2s

˙ˆx1 = M ˆαrz1− ˆαrˆx1− ω ˆx2+ G x2s

˙ˆx2 = M ˆαrz2+ ω ˆx1− ˆαrˆx2+ G x2s,

(36)

where ˆαr= αr rand ˆRλ = Rs+ M µ ˆαr, in which

ˆαr= ˆRr

Lr

= Rr

Lr

where ˆx iandˆz j are the estimation of x i and z j for i ∈ {1, 2}

and j ∈ {1, 2} and G x1 , Gx2 , Gz1 and G z2are the observer

gains

The switching vector  sis chosen as

s =



sign (s1 )

sign (s2 )



with

S =



s1

s2



=



z1− ˆz1

z2− ˆz2



=



ez1

ez2



Setting e x i = x i − ˆx i and e z j = z j − ˆz j for i ∈ {1, 2} and

j ∈ {1, 2}, and using Eqs (35) and (36), the estimation error

dynamics are













ε ˙ez1=µ(+αrex1+ωe x2 r(Mz1− ˆx1)

−G z1s

ε ˙ez2=µ(−ωex1+αrex2 r(Mz2− ˆx2)

−G z2s

˙e x1=−(+αrex1+ωe x2 r(Mz1− ˆx1)

− G x2s

˙e x2=−(−ωex1+ αrex2 r(Mz2− ˆx2)

− G x2s

(40)

Equation (40) can be expressed in a matrix form as







ε ˙ez = µ(αrI − ωJ )ex r(Mz − ˆx)

− G zs

˙e x = −(αrI − ωJ )ex r(Mz − ˆx)

− G x s , (41) where I is the (2 × 2) identity matrix and J is the (2 × 2)

skew symmetric matrix defined by

J =



0 −1



.

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

system of Eqs (35) and (36), e z = (e z1, ez2)T are the fast

variables and e x = (e x1, ex2)Tare the slow variables There-fore, the stability analysis of the above system involves the

determination of G z1 and G z2to ensure the attractiveness of

the sliding surface S(τ ) = 0 in the fast time-scale Thereaf-ter G x1 and G x2are determined, such that the reduced-order

system obtained when S(τ ) ∼ = ˙S(τ) ∼= 0 is locally stable

5.4 Fast reduced-order error dynamics From the singular perturbation theory, the fast reduced-order system of the observation errors can be obtained by

introduc-ing the fast time-scale τ = (t − t0 )/ε The system of Eq (41)

gives







d

dτ ez =µ(αrI − ωJ )ex r(Mz − ˆx)

− G zs

d

dτ ex =−ε(αrI −ωJ )ex r(Mz − ˆx)

−ε G x s.

(42)

Considering ε = 0 in the above system, it yields

d

dτ ez = µ(αrI − ωJ )ex r(Mz − ˆx)

− G zs, (43) d

By an appropriate choice of the observer gain terms G z1

and G z2, sliding mode occurs in (43) along the manifold

ez= 0

Proposition 1 Assume that e x1 and e x2 are bounded in this

time (practical assumption) and that ω varies slowly, and

consider the system of (43) with the following observer gains

matrix

Gz=



η1 0

0 η2



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

given by

ST



dS

dτ



In this time-scale dx/dτ = 0 and de x /dτ = 0 So,

ST d

dτ S = S

T

µ

(αrI −ωJ )ex r(Mz − ˆx)

− G zs

,

(47) or

ST d

dτ S = −s1



η1sign(s1)

−µαrex1 + ωe x2 r(Mz1− ˆx1) 

−s2



η2sign(s2)

−µαrex2 − ωe x1 r(Mz2− ˆx2) 

. (48) Taking into account that all states and parameters of the

induction motor are bounded, there exists sufficiently large

positive numbers η1 and η1such that

ST



dS

dτ



< 0.

Thus, (46) is verified with the set defined by the following

inequalities

η1>µ

αrex1 + ωe x2 r(Mz1− ˆx1) 

η2>µ

αrex2 − ωe x1 r(Mz2− ˆx2) . (49)

Once the trajectory reaches the sliding surface S = e z=

0, the system in the sliding mode behaves as if G zs is

re-placed by its equivalent value (G zs )eq, which can be

calcu-lated from the subsystem (43) assuming e z = 0 and ˙e z= 0

5.5 Slow reduced-order error dynamics

For slow error dynamics (when S ≡ 0), we use the system

(41) and set ε = 0 So, we can write

0= µ(αrI − ωJ )ex r(Mz − ˆx)

˙e x = −(αrI − ωJ )ex r(Mz − ˆx)

− G x s (51) From Eq (50), we can obtain the equivalent switching

vector ˜sas

˜ s = µ G−1

(αrI − ωJ )ex r(Mz − ˆx)

In this time-scale, we can replace  s by ˜s in Eq (51) Hence, subsystem (51) can be written as the following system

˙e x = −K(αrI − ωJ )ex r(Mz − ˆx)

with

in which we assume that G xis a diagonal matrix such that

5.6 Stability analysis of the slow reduced-order error dynamics

We chose the positive-definite candidate Lyapunov function

as follows

2

(ex )Tex+ 1

2

where q > 0.

The t time-derivative of W can be expressed as

˙

W = −kαr(ex )Tex

r

1

q

d

dt r+ k(e x ) T (Mz − ˆx)

The condition for (57) to be negative-definite will be sat-isfied if

and 1

q

d

With the assumption of Eq (59), it yields d

Equation (60) provides an adaptive law to estimate the value

of the rotor resistance Unfortunately, the flux errors (e x ) are

not available So, by defining the function

and by using Eq (41), we obtain

It is possible to reconstruct the estimated fluxes error as

ex= 1

Using the fluxes error estimation of Eq (63), the adaptive law of Eq (60) becomes feasible:

d

dt r= −q

µ k (E − ε ez)

Table 1 Nominal parameters of the induction motor

Ls= 0.274 H Lr= 0.274 H M = 0.258 H

Fig 1 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 actual (solid) speed, c rotor fluxes estimation: ˆ φ rd (solid)

and ˆφ rq (dotted), d rotor fluxes error: eφ rd (solid) and eφ rq (dotted), e reference (dotted) and estimated (solid) rotor resistance, f estimated load

torque

6 Simulation results

The proposed estimation algorithm has been simulated for the

induction motor whose data are given in Table 1 As a

control-ler, the indirect field oriented sliding mode control is used It

is assumed that the load torque is unknown and that all the

parameters are known and constant except for the rotor

resis-tance which will change during the operating motor For this closed loop system, the rotor flux feedback signal and rotor resistance are replaced with the estimated corresponding val-ues of Eqs (36) and (64), respectively With the assump-tion that all states including rotor flux and all parameters are known, the rotor flux and rotor resistance estimated by the proposed method are compared to their actual values

Fig 2 Sensitivity of the system performance to change in the external load with Rr= 1.25R rn and φ∗rd = 1.0 Wb a Real (dotted) and estimated

(solid) load torque, b reference (dotted) and actual (solid) speed, c load torque (dotted) and motor torque (solid), d reference (dotted) and estimated (solid) rotor resistance, e rotor fluxes estimation: ˆ φ rd (solid) and ˆ φ rq (dotted), f rotor fluxes error: eφ rd (solid) and eφ rq (dotted)

The sliding mode control and observer parameters were

chosen as

λω = 120, λ φ = 120, K ω = 80, K φ = 80,

δ1 = δ ω = 0.5, δ2= δ φ = 0.5,

Gz = diag(50, 50), G x = diag(5, 5) and q = 700.

The results are summarized in this section

6.1 Rotor resistance variation effect This test involves in increasing the rotor resistance As shown

in Fig 1a, the motor is started with its nominal rotor

resis-tance value Rrn = 3.805  Then, the rotor resistance of

the motor model is suddenly set to 1.5Rrn at t = 1 s, and

to 2Rrn at t = 2 s The reference speed and reference rotor

flux are maintained at 1400 rpm and 1.0 Wb, respectively

Fig 3 Sensitivity of the system performance to change in the reference speed with Rr= 1.25R rn and φ rd∗ = 1.0 Wb a Reference (dotted) and

actual (solid) speed, b reference (dotted) and estimated (solid) rotor resistance, c estimated rotor fluxes: ˆ φ rd (solid) and ˆ φ rq (dotted), d rotor fluxes

error: eφ rd (solid) and eφ rq (dotted)

Figure 1b shows the speed response of the motor; a very good

speed regulation is obtained Fig 1c, d shows the estimated

rotor fluxes and the error between them and the actual

val-ues High flux tracking and good rotor flux orientation can be

observed Figure 1e compares the estimated and actual rotor

resistance After a short convergence time, the estimated rotor

resistance reaches the actual value Figure 1f shows the

esti-mated load torque These results show that the sliding mode

control with the proposed observer can track the reference

command accurately and quickly It is important to notice

that the q-axis rotor flux error is greater than the d-axis rotor

flux error in the transient state This report is very clear since

the rotor resistance estimation error (in transient state)

prop-agates on the slip frequency which directly affects the rotor

field orientation [see Eq (24)]

6.2 Performance under external load disturbances

The sensitivity of the observer to external load disturbances

is also investigated in this study The objective is to follow

the speed and rotor flux references in spite of disturbances

in the load torque with a constant error (of +25%) in the

rotor resistance value This practical error is made to test the

efficacy of the adaptive law of Eq (64) Figure 2a shows the

actual and the estimated applied load torque Due to the rotor

inertia, the estimated load torque presents negative values in the start-up motor and later follows exactly the actual signal Figure 2b presents a very good performance for speed regu-lation Figure 2c shows the motor and the real load torque Figure 2d presents the actual and estimated rotor resistance Figure 2e, 2f show that the completely decoupled control of rotor flux and torque is obtained and that the observer is very robust to external load disturbances

6.3 Performance over wide speed range

In this case, we consider the speed tracking performances for a wide variation range of reference speed The rotor flux reference is kept at its rated value of 1.0 Wb and the motor operates without external load disturbances The observer performance for speed tracking is presented in Fig 3a The actual and the estimated rotor resistances are shown in Fig 3b

At very low speed, the rotor resistance error is in the order

of 0.7% On the other hand, the rotor resistance estimation is very good at a high speed Figure 3d, 3c show the estimation

of the rotor fluxes and the error between the estimated rotor fluxes and the actual rotor fluxes, respectively These results prove that the speed tracking is quite good and that the rotor field is always well-oriented