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Vieira-2014-Sensorless Sliding-M

Sensorless Sliding-Mode Rotor Speed Observer of Induction Machines Based on Magnetizing

Current Estimation

Rodrigo Padilha Vieira,Member,IEEE,Cristiane Cauduro Gastaldini,

Rodrigo Zelir Azzolin,and Hilton Abílio Gründling

Abstract—This paper presents a rotor speed observer for induction-machine drives based on a sliding-mode approach and magnetizing current estimation.The back electromotive force (EMF)is calculated from the stator current and the stator voltage signals.The magnetizing currents are obtained from the back EMF.A theorem which gives the rotor speed estimate in the continuous-time domain is formulated using the magnetizing cur-rent estimation.The stability analysis is achieved based on the Lyapunov approach.Moreover,the discrete-time approach of the method is discussed,and a theorem for the discrete-time imple-mentation is proposed.The limits that ensure the system stability for the switching gains and for the observer gain are discussed. This paper shows that the limits for the gains of the discrete-time algorithm are different from the limits for the gains of the continuous-time algorithm.Simulation results are presented to validate the theoretical analysis.In addition,experimental results based on a?xed-point digital signal processor(DSP)platform (DSP TMS320F2812)demonstrate the performance of the pro-posed scheme.

Index Terms—Induction machine(IM),sensorless control, sliding-mode observer,variable-speed drive.

I.I NTRODUCTION

I NDUCTION machines are widely used in the industrial ap-

plications for electromechanical conversion due to their ro-bustness,cost,and simplicity.The?eld-oriented control(FOC) methods improve the performance of the induction machine (IM)drive and have been well established in the literature, such as in[1]–[4].The FOC methods require the information of IM state variables,i.e.,electromagnetic torque,rotor speed, or dq stator currents.As a consequence,the state estimation problem has been extensively studied in the literature.The IM drives without mechanical sensors are commonly called

Manuscript received March28,2013;revised June11,2013and August20, 2013;accepted September23,2013.Date of publication November20,2013; date of current version March21,2014.This work was supported in part by the Coordination for the Improvement of the Higher Level Personnel(CAPES)and in part by the Brazilian Government.

R.P.Vieira and H.A.Gründling are with the Power Electronics and Control Research Group(GEPOC),Federal University of Santa Maria(UFSM), 97105-900Santa Maria,Brazil(e-mail:rodrigovie@https://www.sodocs.net/doc/c03814528.html,;ghilton03@ https://www.sodocs.net/doc/c03814528.html,).

C.C.Gastaldini is with the Federal University of Pampa(UNIPAMPA), 96413-170Bagé,Brazil(e-mail:gastaldini@https://www.sodocs.net/doc/c03814528.html,).

R.Z.Azzolin is with the Federal University of Rio Grande(FURG),96203-900Rio Grande,Brazil(e-mail:rodrigoazzolin@https://www.sodocs.net/doc/c03814528.html,).

Color versions of one or more of the?gures in this paper are available online at https://www.sodocs.net/doc/c03814528.html,.

Digital Object Identi?er10.1109/TIE.2013.2290759“sensorless”systems,which use a scheme to estimate the rotor speed.The most popular rotor speed estimation methods are IM model based,such as the extended Kalman?lter,model reference adaptive systems,and sliding-mode techniques[4]–[9].In addition,several papers in the literature present robust, adaptive,and decoupled control strategies applied to electrical machine drives[10],[11].

The sliding-mode methods are characterized by their sim-plicity of implementation,disturbance rejection,and strong ro-bustness.These schemes are used for control and estimation in several processes,such as those in[12]–[22].The conventional approach of the sliding-mode scheme uses a high-frequency switched control law dependent on the system state.Utkin[7] presents a variety of applications in electric drives of the sliding-mode methods,including rotor speed control of IM drives.In[23],an observer aiming to estimate the rotor speed and the rotor resistance is proposed.The algorithm is based on the estimation of stator currents using a continuous-time sliding-mode function.Rao et al.[13]present a scheme for the estimation of rotor speed,rotor?ux vector,and rotor resistance. The method uses two switching surfaces based on the stator current estimation error and the observed rotor?ux.Results showed the good performance of the method.

On the other hand,some studies have developed discrete-time sliding-mode approaches aiming the digital imple-mentation.In this case,the stability analysis achieved in the continuous-time domain cannot be directly extended to discrete-time systems.The sampling time of the processor results in the chattering phenomenon due to the fact that the sampling frequency is limited to the processor capability.One solution to cope with this problem is the formulation and stabil-ity analysis achieved in discrete time.This design approach has been attractive to several researchers owing to the implemen-tation easiness on digital signal processors(DSPs)and micro-controllers[24]–[30].

Vieira et al.[28]present a discrete-time sliding-mode ro-tor speed observer applied to induction motor drives.The sliding-mode observer is based on the stator current estimation. Differently from the previous rotor speed observer schemes based on the sliding-mode approaches,the method proposed in[28]presents the algorithm stability analysis on the discrete-time domain.The conditions for the existence of the discrete-time sliding surfaces are presented,and an algorithm for rotor speed estimation is proposed.In this paper,we propose a rotor

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speed observer based on the estimation of magnetizing currents. The back electromotive force(EMF)vector is calculated from the stator current and the stator voltage signals.Thus,it is possible to compute the magnetizing currents using the back-EMF vector.Initially,a continuous-time algorithm is proposed, and its performance is evaluated through simulation results. In addition,we formulate the stability analysis problem in the discrete-time domain.A discrete-time rotor speed estimation algorithm is proposed.The main contribution of this paper is the formulation and the analysis of two schemes which provide the rotor speed of IM drives in continuous-time and discrete-time domains.The stability analysis is achieved for continuous-time and discrete-time methods using the Lyapunov approach.The formulation in discrete time allows to obtain the relationship between the sampling time and the observer gain,aiming to ensure the system stability.Moreover,the conditions for the discrete-time sliding-mode switching hyperplane existence are presented.Simulation and experimental results are presented to validate the theoretical analysis and the performance of the proposed scheme.

This paper is organized as follows.Section II presents the IM mathematical model.Section III gives the continuous-time rotor speed observer.Section IV proposes a discrete-time rotor speed observer.Section V shows the simulations results while Section VI presents the experimental results.Section VII presents the conclusions of this paper.

II.IM M ODEL

The behavior of the IM can be modeled by continuous-time differential equations in a stationary reference frame(qd s), such as[31]

i sq=?γi sq+β

r φrq?βpωrφrd+

v sq(1)

d dt i sd=?γi sd+βpωrφrq+β

τr

φrd+

σL s

v sd(2)

d dt φrq=?

τr

φrq+pωrφrd+

τr

L m i sq(3)

d dt φrd=?

τr

φrd?pωrφrq+

τr

L m i sd(4) T e=

L m

L r

p(φrd i sq?φrq i sd)(5)

d dt ωr=?

B n

ωr+

(T e?T L)(6)

where R s and R r are the stator and rotor resistances,L s and L r are the stator and rotor inductances,and L m is the mutual inductance;i sq,i sd,φrq,φrd,v sq,and v sd are the stator currents,the rotor?uxes,and the stator voltages,respectively,ωr is the rotor speed,T e is the electromagnetic torque,T L is the load torque,J is the moment of inertia,B n is the friction coef?cient,and p is the pole pair number.The constants in the aforementioned expressions are de?ned as

τrΔ=L r

R r

,σΔ=1?

L2m

L s L r

,βΔ=

L m

σL s L r

,γΔ=

R s

σL s

+β

τr

L m.

The expressions of back EMF can be calculated from the

current and voltage signals as

e mq=v sq?R s i sq?σL s

i sq(7)

e md=v sd?R s i sd?σL s

i sd.(8)

It is possible to obtain the back-EMF equations from the

magnetizing currents in the form

e mq=L m

i qM(9)

e md=L m

i dM(10)

where L m=L2m/L r,and the magnetizing currents could be

given by

i qM=

L r

L m

i rq+i sq(11)

i dM=

L r

L m

i rd+i sd(12)

where i rq and i rd are the rotor currents.

The differential equations of magnetizing currents can be

given by

i qM=?

τr

i qM+pωr i dM+

τr

i sq(13)

i dM=?

τr

i dM?pωr i qM+

τr

i sd.(14)

Furthermore,the differential equations of magnetizing cur-

rents also can be obtained from the back EMF(9)and(10),

such as

i qM=e mq/L m(15)

i dM=e md/L m.(16)

Thus,from(15)and(16),it is possible to compute the

magnetizing currents using the calculated back EMF.

The expressions(13)and(14)and(15)and(16)present two

methods to obtain the magnetizing currents.The?rst method

uses the stator currents and a component that include the

rotor speed information,while the second method calculates

the magnetizing currents directly from the back EMF.The

?rst method cannot be implemented without the rotor speed

information.The second method uses only voltage and current

signals.As a consequence,an observer based on the sliding-

mode approach for the magnetizing currents can be used,aim-

ing to obtain the rotor speed information.Moreover,the rotor

speed information is associated with the magnetizing current

[see(13)and(14)];then,a second observer is used to estimate

the rotor speed as depicted in the next section.

VIEIRA et al.:SENSORLESS SLIDING-MODE ROTOR SPEED OBSERVER OF INDUCTION MACHINES4575

III.C ONTINUOUS-T IME R OTOR S PEED O BSERVER

A.Magnetizing Current Estimation

A sliding-mode observer for the magnetizing current can be

designed as

d dt ?i

τr

i sq+Uα(17)

i dM=?

r ?i

i sd+Uβ(18)

where Uαand Uβare discontinuous functions given by

Uα=?U0sign(ˉi qM)(19)

Uβ=?U0sign(ˉi dM)(20) with U0∈ +.

Thus,the sliding surfaces are

ˉi

=?i qM?i qM(21)

ˉi

=?i dM?i dM.(22) Lemma1:Consider the sliding surfacesˉi qM andˉi dM pre-sented in(21)and(22),the expressions for Uαand Uβgiven in (19)and(20).Then,for U0∈ +and large enough estimates of?i qM and?i dM,track the computed values of i qM and i dM, respectively.

Proof:The differential equations of the magnetizing cur-rent estimation errors are obtained from(13),(14),(17),and (18),such as

d dt ˉi

τr

ˉi

+Uα?pωr i dM(23)

d dt ˉi

τr

ˉi

+Uβ+pωr i qM.(24)

Thus,a Lyapunov candidate function can be written by

V=1

ˉi2

+ˉi2dM

.(25)

The derivative of(25)is obtained such as

˙V=ˉi

qM ˙ˉi

+ˉi dM˙ˉi dM.(26)

Using(23)and(24)in(26)results in

˙V=?U

0(|ˉi qM|+|ˉi dM|)?

τr

ˉi2

+ˉi2dM

+pωr(i qMˉi dM?i dMˉi qM)(27)

which means that,for a large enough U0,then,˙V<0,and the sliding mode will occur in the surfacesˉi qM=0andˉi dM=0. When the sliding mode occurs,ˉi qM=0,andˉi dM=0, then,the sliding-mode dynamics can be obtained replacing the discontinuous functions Uαand Uβby their equivalent control components Uαeq and Uβeq,whose calculated settings are(d/dt)ˉi qM,ˉi qM,(d/dt)ˉi dM,andˉi dM to0in(23)and(24) [7].Thus,

Uαeq=pωr i dM(28)

Uβeq=?pωr i qM(29)where Uαeq and Uβeq can be obtained from the discontinuous functions Uαand Uβusing low-pass?lters,as presented in[7] and[32].

Note that the information of the rotor speed could be obtained from(28)and(29)since the sliding surface occurs inˉi qM and ˉi

;however,i qM and i dM have a sinusoidal behavior,and the numerical solution could result in division by zero.

B.Rotor Speed Observer

Consider the following assumption.

A1:The dynamics of the mechanical variables,such as rotor speed,are more slower than the dynamics of the electrical variables such as stator currents and rotor?uxes;then,it is reasonable to assume˙ωr=0.

Remark1:Assumption A1is not very restrictive.Equations (1)–(4)present the dynamics of electrical variables,while (6)presents the differential equation of the rotor speed.We can observe that the rotor speed dynamic is affected by the electromagnetic torque and load torque;moreover,the rotor speed variable depends on B n and https://www.sodocs.net/doc/c03814528.html,monly,the dynamics associated to the coef?cients B n and J are more slower than the electrical dynamics;as a result,the rotor speed dynamic has behavior similar to that of a low-pass?lter of the electromag-netic torque,and the rotor speed could be assumed reasonably constant in comparison with the variation of the electrical variables[23],[32],[33].

From A1,the derivatives of(28)and(29)can be written in the form

Uαeq=?

Uαeq+pωr Uβeq+pωr

i sd(30)

Uβeq=?

τr

Uβeq?pωr Uαeq?pωr

τr

i sq.(31)

An observer for(30)and(31)can be designed as

αeq

τr

Uαeq+p?ωr Uβeq+p?ωr

τr

i sd?K(?Uαeq?Uαeq)

(32) d

βeq

=?p?ωr Uαeq?

τr

Uβeq?p?ωr

τr

i sq?K(?Uβeq?Uβeq)

(33) where K is a positive gain.

The estimative errors areˉUαeq=?Uαeq?Uαeq andˉUβeq=?U

βeq?Uβeq,and their derivatives are written as

ˉU

αeq

=?KˉUαeq+pˉωr Uβeq+pˉωr

τr

i sd(34)

ˉU

βeq

=?KˉUβeq?pˉωr Uαeq?pˉωr

τr

i sq(35)

whereˉωr=?ωr?ωr.

Theorem1:Consider the sliding surfacesˉi qM andˉi dM,the assumption A1,and the observer given in(32)and(33).Then, the adaptation law,given by

˙ˉω

Δ=?pU

βeq

ˉU

αeq?p

τr

i sdˉUαeq+pUαeqˉUβeq+p

τr

i sqˉUβeq

(36) is stable and ensures the convergence of?ωr toωr,as t→∞.

4576IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.61,NO.9,SEPTEMBER

2014

Fig.1.Rotor speed observer block diagram.

Proof:Consider the Lyapunov candidate function

V=1

U2αeq+ˉU2βeq+ˉω2r

≥0.(37)

The derivative of(37)is

˙V=ˉU

αeq ˙ˉU

αeq

+ˉUβeq˙ˉUβeq+ˉωr˙ˉωr.(38)

Replacing(34)and(35)in(38)results in

˙V=?K ˉ

U2αeq+ˉU2βeq

+pˉωr UβeqˉUαeq+pˉωr

τr

i sdˉUαeq

?pˉωr UαeqˉUβeq?pˉωr1

τr

i sqˉUβeq+ˉωr˙ˉωr.(39)

Using the adaptive law(36)in(39)

˙V=?K ˉ

U2αeq+ˉU2βeq

.(40)

Thus,from(40),it is possible to conclude that K>0under the adaptation law(36),the function(38)is negative,and the variablesˉUαeq andˉUβeq are close to zero when t→∞. It is possible to write˙ˉωr=˙?ωr from assumption A1;as a consequence,the estimated rotor speed(?ωr)is calculated using the integral of˙?ωr.

One of the advantages of the magnetizing current estimation is to avoid the estimation of the rotor?ux vector.In addition, the in?uence of noise in measured signals,such as the stator currents,can be reduced by the computation of the voltage signal in the back-EMF expressions(7)and(8).In some prac-tical applications,the voltage signal can be obtained from the modulation index and the nominal dc bus value,avoiding noise and offset in measured voltage signals.The block diagram of the proposed continuous-time algorithm is presented in Fig.1.

IV.D ISCRETE-T IME R OTOR S PEED O BSERVER

A.Discretized Model

The differential equations of the IM dynamics given in (1)–(4)can be discretized by the Euler method choosing a sampling time T s as

i sq(k+1)=(1?γT s)i sq(k)+β1

τr

T sφrq(k)

?βpωr(k)T sφrd(k)+1

σL s

T s V sq(k)(41) i sd(k+1)=(1?γT s)i sd(k)+βpωr(k)T sφrq(k)

+β1

τr

T sφrd(k)+

σL s

T s V sd(k)(42)

φrq(k+1)=

τr

T s

φrq(k)+pωr(k)T sφrd(k)

τr

L m T s i sq(k)(43)

φrd(k+1)=

τr

T s

φrd(k)?pωr(k)T sφrq(k)

τr

L m T s i sd(k).(44)

The discretized back-EMF expressions are given by

e mq(k)=V sq(k)?R s i sq(k)?σL s

Δi sq(k)

T s

(45)

e md(k)=V sd(k)?R s i sd(k)?σL s

Δi sd(k)

T s

.(46)

Moreover,the back-EMF expressions can be written as

e mq(k)=L m

Δi qM(k)

T s

(47)

e md(k)=L m

Δi dM(k)

T s

(48)

whereΔi qM(k)=i qM(k+1)?i qM(k)andΔi dM(k)=i dM(k+1)?

i dM(k).

The expressions of magnetizing currents are also discretized

using the Euler method,such as

i qM(k+1)=

τr

T s

i qM(k)+pωr(k)T s i dM(k)+

τr

T s i sq(k)

(49)

i dM(k+1)=

τr

T s

i dM(k)?pωr(k)T s i qM(k)+

τr

T s i sd(k).

(50)

The magnetizing currents can be written from(47)and(48),

using the back EMF

i qM(k+1)=

L m

e mq(k)T s+i qM(k)(51)

i dM(k+1)=

L m

e md(k)T s+i dM(k).(52)

B.Discrete-Time Magnetizing Current Estimation

A discrete-time sliding-mode magnetizing current observer

can be designed in the form

qM(k+1)

τr

T s

qM(k)

τr

T s i sq(k)+T s Uα(k)(53)

dM(k+1)

τr

T s

dM(k)

τr

T s i sd(k)+T s Uβ(k)(54)

where Uα(k)and Uβ(k)are discontinuous functions of the

magnetizing current estimation error,such as

Uα(k)=?U0αsign

ˉi

qM(k)

=?U0αsign

qM(k)?i qM(k)

(55)

Uβ(k)=?U0βsign

ˉi

dM(k)

=?U0βsign

dM(k)?i dM(k)

(56)

where U0αand U0βare positive constants.

VIEIRA et al.:SENSORLESS SLIDING-MODE ROTOR SPEED OBSERVER OF INDUCTION MACHINES4577 Discrete-Time Sliding Conditions:The conditions for the ex-

istence of the discrete-time sliding-mode surfaces are discussed

in several papers,such as[14]and[34].In the discrete-time

approach,the conditions derived directly from the continuous-

time schemes do not ensure the stable convergence of the

discrete-time approach.The stable convergence in the discrete-

time approach could be obtained from the Lyapunov analysis,

and it could be given by two inequations,such as

s i(k+1)?s i(k)

sign

s i(k)

<0(57)

s i(k+1)+s i(k)

sign

s i(k)

≥0(58)

where s i(k)expresses a generic discrete-time sliding surface. Lemma2:Consider the discrete-time sliding surfacesˉi qM(k) andˉi dM(k)[see(59)and(60)],the discontinuous functions Uα(k)and Uβ(k)given in(55)and(56).Then,it is possible to ensure the existence of positive values for U0αand U0βthat guarantee the convergence of the estimated values of?i qM(k) and?i dM(k)to the calculated values of i qM(k)and i dM(k), respectively.

Proof:The estimation magnetizing current errors are ob-tained from(49),(50),(53),and(54),such as

ˉi

qM(k+1)=

τr

T s

ˉi

qM(k)?pωr(k)T s i dM(k)+T s Uα(k)

(59)

ˉi

dM(k+1)=

τr

T s

ˉi

dM(k)

+T s Uβ(k)+pωr(k)T s i qM(k).

(60)

From(59)and(60),it is possible to obtain the difference equations ofˉi qM(k)andˉi dM(k)

Δˉi qM(k)=ˉi qM(k+1)?ˉi qM(k)=?1

τr

T sˉi qM(k)+T s Uα(k)

?pωr(k)T s i dM(k)(61)

Δˉi dM(k)=ˉi dM(k+1)?ˉi dM(k)=?1

τr

T sˉi dM(k)+T s Uβ(k)

+pωr(k)T s i qM(k).(62) Multiplying(61)and(62)by the signal function of the magnetizing current estimation error,it is possible to obtain

ˉi

qM(k+1)?ˉi

qM(k)

sign

ˉi

qM(k)

=?1

τr

T s

i qM(k)

?T s U0α

?pωr(k)T s i dM(k)sign

ˉi

qM(k)

(63)

ˉi

dM(k+1)?ˉi

dM(k)

sign

ˉi

dM(k)

=?1

τr

T s

i dM(k)

?T s U0β

+pωr(k)T s i qM(k)sign

ˉi

dM(k)

.(64)

From(63)and(64),it is possible to verify that the suitable

choice of U0αand U0βbeing large enough ensures the nec-

essary condition for the existence of the discrete-time sliding

surface presented in(57).Thus

U0α>?

τr

i qM(k)

?pωr(k)i dM(k)sign

ˉi

qM(k)

(65)

U0β>?

τr

i dM(k)

+pωr(k)i qM(k)sign

ˉi

dM(k)

.(66)

Aiming to obtain the suf?cient condition given in(58),it is

possible from(59)and(60)to write the following expressions:

ˉi

qM(k+1)

+ˉi qM(k)

sign

ˉi

qM(k)

τr

T s

i qM(k)

?T s U0α

?pωr(k)T s i dM(k)sign

ˉi

qM(k)

i qM(k)

(67)

ˉi

dM(k+1)

+ˉi dM(k)

sign

ˉi

dM(k)

τr

T s

i dM(k)

?T s U0β

+pωr(k)T s i qM(k)sign

ˉi

dM(k)

i dM(k)

.(68)

Thus,the upper limit for U0αand U0βis de?ned from(67)

and(68),such as

U0α≤

i qM(k)

T s

τr

i qM(k)

?pωr(k)i dM(k)sign

ˉi

qM(k)

(69)

U0β≤

i dM(k)

T s

τr

i dM(k)

+pωr(k)i qM(k)sign

ˉi

dM(k)

(70)

C.Discrete-Time Rotor Speed Observer

Assuming that the discrete-time sliding surfaces exist,the

calculated back-EMF values are true,and the estimated magne-

tizing currents track the calculated magnetizing currents.Then,

the discrete-time sliding-mode dynamics can be obtained by

replacing the discontinuous functions Uα(k)and Uβ(k)by their

equivalent control components Uαeq(k)and Uβeq(k),which are

calculated de?ningˉi qM(k+1)=0,ˉi qM(k)=0,ˉi dM(k+1)=0,

andˉi dM(k)=0in(59)and(60).Thus,

Uαeq(k)=pωr(k)i dM(k)(71)

Uβeq(k)=?pωr(k)i qM(k).(72)

The variables Uαeq(k)and Uβeq(k)can be obtained from

the discontinuous functions Uα(k)and Uβ(k)using low-pass

?lters[7].

Here,the following assumption is made.

A2:It is possible to assume that,for small values of T s,the

variation of the mechanical rotor speed over one sampling time

is slower than the variation of the electrical variables such as

stator currents and back EMF.Then,the mechanical rotor speed

could be considered constant over one sampling time,such as

ωr(k+1))≈ωr(k).

4578IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.61,NO.9,SEPTEMBER 2014

Thus,the difference expressions of (71)and (72)from as-sumption A2can be rewritten in the form U αeq (k +1)=U αeq (k )?

τr

T s U αeq (k )+pωr (k +1)T s U βeq (k )+

τr

T s pωr (k +1)i sd (k )(73)U βeq (k +1)

=U βeq (k )?1

τr

T s U βeq (k )?pωr (k +1)T s U αeq (k )

τr

T s pωr (k +1)i sq (k ).(74)A discrete-time parameter observer can be designed for the

system presented in (73)and (74),such as

αeq (k +1)=(1?KT s )?U αeq (k )?1τr

T s U αeq (k )+p ?ωr (k +1)T s U βeq (k )+KT s U αeq (k )

τr

T s p ?ωr (k +1)i sd (k )(75)?U βeq (k +1)=(1?KT s )?U βeq (k )?p ?ωr (k +1)T s U αeq (k )

τr T s U βeq (k )+KT s U βeq (k )

τr T s p ?ωr (k +1)i sq (k ).(76)The expressions of estimation errors of U αeq (k +1)and U βeq (k +1)are given by

ˉU αeq (k +1)=(1?KT s )ˉU αeq (k )+p ˉωr (k +1)T s U βeq (k )

+1τr

p ˉωr (k +1)T s i sd (k )(77)ˉU βeq (k +1)=(1?KT s )ˉU βeq (k )?p ˉωr (k +1)T s U αeq (k )

?1τr p ˉωr (k +1)T s i sq (k )(78)where ˉU

αeq (k )=?U αeq (k )?U αeq (k ),ˉU βeq (k )=?U βeq (k )?U βeq (k ),and ˉω

r (k )=?ωr (k )?ωr (k ).Theorem 2:Consider the discrete-time sliding surfaces ˉi qM (k )and ˉ

i dM (k ),the parameter observer presented in (75)and (76)under assumption A2.Then,for λ∈ +and KT s ∈(0,1],the error adaptation algorithm given by (79),shown at the bottom of the page,ensures the stable convergence of the ?ωr (k )to ωr (k )when k →∞.

The variables presented in (79)are given by U αβ(k ) 2=(U 2βeq (k )+U 2αeq (k ))and i qd (k ) 2=(i 2sd (k )+i 2sq (k )).

Proof:Consider the Lyapunov candidate function

V k =ˉU 2αeq (k )+ˉU 2βeq (k )+λ?1ˉω

2r (k ).(80)

The difference equation of (80)is given by ΔV k =ˉU 2αeq (k +1)?ˉU 2αeq (k )+ˉU 2βeq (k +1)?ˉU 2βeq (k )

+λ?1ˉω2r (k +1)?λ?1ˉω

rk .(81)

It is possible to write

ω2r (k +1)?ω2rk Δ

=2ωr (k +1)Δωrk ?Δω2

(82)

where Δωrk =ωr (k +1)?ωr (k ).

Replacing (77),(78),and (82)in (81)results in

ΔV k =? 1?(1?KT s )2 ˉU 2αeq (k )+ˉU 2βeq (k )

+p 2ˉω2r (k +1)T 2s U 2βeq (k )+U 2αeq (k )

+1τ2r

p 2ˉω2r (k +1)T 2s i 2sd (k )+i 2

sq (k ) +2(1?KT s )p ˉωr (k +1)T s

×(ˉU

αeq (k )U βeq (k )?ˉU βeq (k )U αeq (k ))+2(1?KT s )1τr

p ˉωr (k +1)T s

ˉU

αeq (k )i sd (k )?ˉU βeq (k )i sq (k ) +21τr p 2ˉω2r (k +1)T 2s

U βeq (k )i sd (k )+U αeq (k )i sq (k ) +λ?12ωr (k +1)Δωrk ?λ?1Δω2rk .

(83)

From the adaptation law (79),it is possible to write

Δˉωr (k )=?12

λp 2ˉωr (k +1)T 2s U 2βeq (k )+U 2

αeq (k )

?1λ1r

p 2ˉωr (k +1)T 2s i 2sd (k )+i 2

sq (k ) ?λ(1?KT s )pT s ˉU αeq (k )U βeq (k )?ˉU βeq (k )U αeq (k ) ?λ(1?KT s )1τr

pT s ˉU αeq (k )i sd (k )?ˉU βeq (k )i sq (k ) ?λ1τr

p 2ˉω

r (k +1)T 2

s U βeq (k )i sd (k )+U αeq (k )i sq (k ) .(84)Replacing (84)in (83)results in

ΔV k = (1?KT s )2?1 ˉU 2

αeq (k )+ˉU 2βeq (k ) ?λ?1Δω2r (k ).(85)For values of KT s ∈(0,1],the function (81)is negative.

Then,the variables ˉU

αeq (k )and ˉU βeq (k )converge to zero when k →∞.

ˉωr (k +1)=

ˉωr (k )

1+λp 2T 2s 12

U αβ(k ) 2

121τ2r

i qd (k ) 2+1τr U βeq (k )i sd (k )+U αeq (k )i sq (k ) ?λ(1?KT s )pT s ˉU αeq (k )U βeq (k )?ˉU βeq (k )U αeq (k ) +1τr ˉU αeq (k )i sd (k )?ˉU βeq (k )i sq (k ) 1+λp 2T 2s 12 U αβ(k ) 2+121τ2r

i qd (k ) 2+1τr U βeq (k )i sd (k )+U αeq (k )i sq (k )

(79)

VIEIRA et al.:SENSORLESS SLIDING-MODE ROTOR SPEED OBSERVER OF INDUCTION MACHINES

4579

Fig.2.Control system diagram.

TABLE I IM P

ARAMETERS

By assumption A2,we have Δˉωrk =Δ?ωrk ;as a conse-quence,assuming that the initial condition for ?ωr (k )is known,then,it is possible to compute the estimated rotor speed ?ωr (k +1)as follows:

Δ ωrk = ωr (k +1)? ωr (k )=ωr (k +1)?ωr (k ).

(86)

Note that the adaptation law (79)cannot be obtained directly

from the discretization of (36).The development achieved in discrete time in Section IV allows to establish the relationship between the observer gain and sampling time,which ensures the system stability.

V .S IMULATION R ESULTS

In order to validate the theoretical analysis and to verify the performance of the proposed method,simulation results are obtained.An indirect ?eld-oriented control (IFOC)scheme with a qd e reference frame rotating at synchronous speed ωe is used for IM rotor speed control.The proposed continuous-time scheme is simulated using the Matlab/Simulink and SimPower-Systems Library,as presented in the simpli?ed block diagram of Fig.2.The block diagram of the proposed rotor speed estimation scheme is shown in Fig.1.Proportional–integral-type controllers are used in the rotor speed control loop and in the qd e stator current control loop.The parameters of the simulated IM machine are listed in Table I.The designed algorithm gains are as follows:U 0=400and K =80.In

the ?rst test,the rotor speed reference (ω?

)varies from 0to 100rad/s,at smooth steps.Fig.3presents the rotor speed response of the proposed continuous-time method.It is possible to verify the good convergence of the estimated rotor speed to the actual rotor speed and the tracking of the reference rotor speed.The oscillations on the estimated rotor speed could be reduced by the choice of K and U 0or by the bandwidth choice of the low-pass ?lter in (28)and (29).The choice of a high value for the U 0variable ensures a fast convergence of the estimated to the calculated magnetizing current;however,it can result

Fig.3.Simulated rotor speed

response.

Fig.4.Calculated and estimated magnetizing currents.(a)i qM and ?i qM .

(b)i dM and ?i dM .

oscillations on the estimated rotor speed.On the other hand,

the choice of the K gain has effect on the convergence of the estimated to the actual rotor speed.

Fig.4presents the calculated and estimated magnetizing currents from the back EMF.Fig.4(a)and (b)illustrates the estimation of the magnetizing currents for the q -and d -axes.Fig.5shows the ?ltered U αeq and U βeq variables and observed ?U

αeq and ?U βeq .The ?ltered U αeq and U βeq variables present a high-frequency component due the switching of the sliding-mode law.These high-frequency components could be reduced with the choice of the low-pass ?lter bandwidth in (28)and (29).However,these high-frequency components in the U αeq and U βeq variables do not degrade the rotor speed estimation.Simulation results with parameter variations are obtained to verify the in?uence of parameter uncertainties.Fig.6presents the speed response with parameter variations.In the instant of 3s,the rotor resistance is changed to 1.5p.u.of the rated value.We can observe additional oscillations on the estimated rotor speed;however,the average of the estimated rotor speed converges to the actual rotor speed.The parameters of the algorithm in this simulation are U 0=350[see (19)and (20)]and K =60[see (32)and (33)].

4580IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,VOL.61,NO.9,SEPTEMBER

2014

Fig.5.Calculated and observed U αβeq variables.(a)U αeq and ?U

αeq .(b)U βeq and ?U

βeq

.Fig.6.Simulated rotor speed response with parameter variation.

VI.E XPERIMENTAL R ESULTS

The performance of the proposed discrete-time scheme pre-sented in Section IV is evaluated on an experimental setup built

with a TMS320F2812DSP,a voltage source inverter (VSI),and an IM.The simpli?ed block diagram of the implemented system is presented in Fig.2.An IFOC scheme is used.The IM parameters are presented in Table I.The switching frequency adopted was 5kHz,K =1550,and λ=1.2.The sampling period selected was 200μs,aiming to reduce the switching losses at VSI and to allow the execution of the proposed control algorithm on the ?xed-point DSP.

The DSP implementation is carried out by the direct compu-tation of discrete-time equations given in Section IV.The ?rst step is the measurement of the stator currents,by Hall effect sensors,and the acquisition of the stator voltage signals.Here,the voltage signals are obtained by the modulation index and by the nominal dc-link value.Thus,the back-EMF vector can be calculated by expressions (45)and (46);as a consequence,the magnetizing currents are computed by (51)and (52).The initial conditions are assumed zero.The i sq (k +1)and i sd (k +1)currents can be obtained by ?lters,such as the state variable ?lter also used in [35].The sliding-mode magnetizing current observer is calculated by (53)and (54).The sliding-mode equivalent terms U αeq (k )and U βeq (k )are used in the observer (75)and

(76).

Fig.7.Experimental rotor speed response.

Then,the estimated rotor speed can be directly obtained from

(79)and (86).

The discrete-time sliding-mode approaches are character-ized by the quasi-sliding-mode band around the switching hyperplane.The bounds for the existence of the discrete-time hyperplane depend on the switching gains U 0αand U 0βgiven in (65),(66),(69)and (70).These bounds vary with the es-timation error;thus,in order to improve the performance of the scheme,variable switching gains are used.The sigmoid function is adopted as the switching function,aiming to reduce the chattering phenomenon,as carried out in [28].This function is represented as follows:

f (x )=?U si

g ?0.5+

1+e ?τsig x

(87)where τsig is the time constant of the function,U sig is a positive gain similar to U 0αand U 0β,and x is a function of error,such

as ˉi qM or ˉ

i dM .The rotor speed range of interest in this paper varies from approximately 8%to the rated rotor speed.Thus,this speed range is presented by three experiments.The ?rst experiment demonstrated the variation on the rotor speed reference from 0to 100rad/s,at smooth steps.Fig.7shows the speed response of the proposed discrete-time algorithm.This ?gure illustrates the good performance and the good capability of speed estimation of the proposed discrete-time scheme.

Fig.8presents the calculated and estimated magnetizing cur-rents,which are obtained from the calculated back EMF as (51)and (52).These equations depend on the integration of magne-tizing currents signals.The result of this integration can be ob-served in Fig.8(a)and (b),and both have a dc offset.However,this dc offset does not degrade the rotor speed estimation due to the fact that the rotor speed depends on frequency signals.In addition,the integration of measured signals in the presence of noise and offset could result in variable over?ow.One solution to compensate the offset in the measured signals is presented in [36].Here,to avoid the integration of the magnetizing currents,a limit plan majored by the measured stator currents is used.This limit plan can be implemented,i.e.,for i qM ,as

i qM =?

??i qM ,if |i qM |≤ i 2sq +i 2sd

i qM |i qM |

i 2sq +i 2sd ,if |i qM |> i 2sq +i 2sd .

(88)

VIEIRA et al.:SENSORLESS SLIDING-MODE ROTOR SPEED OBSERVER OF INDUCTION MACHINES

4581

Fig.8.Calculated and estimated magnetizing currents for the experimental

test.(a)i qM and ?

i qM .(b)i dM and ?i dM

.Fig.9.Experimental rotor speed response with reverse

operation.

Fig.10.Calculated and observed U αβeq variables for the second experimen-tal test.(a)U αeq and ?U

αeq .(b)U βeq and ?U βeq .Fig.9gives the rotor speed response for the second experi-ment.In the second experiment,the aim is to verify the reverse

speed operation and the zero crossing.This ?gure demonstrates the good capacity of estimation of the speed for the tested entire range of operation.Fig.10shows the calculated and

observed

Fig.11.Experimental rotor speed response under load variation.

values for U αeq and U βeq variables.We can observe the good

estimation of these variables.

Fig.11presents the third experiment.This experiment gives the operation with load variation.Between 5and 9s,a load torque is applied at the mechanical axis by means of a dc generator,as can be seen in Fig.11.This ?gure indicates that the proposed discrete-time method has a good speed response even in the presence of a load disturbance.

VII.C ONCLUSION

This paper has developed a scheme for rotor speed estimation for induction motor drives.The proposed method calculates and estimates the magnetizing currents from the back EMF.The convergence of the proposed continuous-time and discrete-time algorithms is analyzed by the Lyapunov approach.This paper presented the switching gain limits that ensure the sta-bility of the discrete-time approach.These limits are related with the time sampling and estimation errors.In addition,this paper presented simulation and experimental results,aiming to validate and to demonstrate the effectiveness of the proposed scheme.The experimental results show zero crossing tests,load disturbance,and speed variation from approximately 8%to the nominal rotor speed.

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2003.

Rodrigo Padilha Vieira(M’13)was born in Cruz

Alta,Brazil,in1983.He received the B.S.degree

in electrical engineering from the Universidade Re-

gional do Noroeste do Estado do Rio Grande do Sul

(Unijuí),Ijuí,Brazil,in2007and the M.S.and

Dr.Eng.degrees in electrical engineering from the

Federal University of Santa Maria(UFSM),Santa

Maria,Brazil,in2008and2012,respectively.

From2010to2014,he was with the Federal

University of Pampa,Alegrete,Brazil.Since2014,

he has been with the UFSM,where he is currently an Adjunct Professor.His research interests include electrical machine drives, sliding-mode control,and digital control techniques for static converters. Dr.Vieira is a member of the IEEE Industrial Electronics

Society.

Cristiane Cauduro Gastaldini was born in Campo

Grande,Brazil,in1984.She received the B.S.,M.S.,

and Dr.Eng.degrees in electrical engineering from

the Federal University of Santa Maria,Santa Maria,

Brazil,in2007,2008,and2012,respectively.

Since2011,she has been with the Federal Univer-

sity of Pampa,Bagé,Brazil,where she is currently

an Adjunct Professor.Her research interests include

electrical machine drives,sliding-mode control,and

digital control techniques for static

converters.

Rodrigo Zelir Azzolin was born in S?o Luiz Gon-

zaga,Brazil.He received the B.S.,M.S.,and Dr.Eng.

degrees in electrical engineering from the Federal

University of Santa Maria,Santa Maria,Brazil,in

2007,2008,and2012,respectively.

Since2010,he has been with the Federal Univer-

sity of Rio Grande,Rio Grande,Brazil,where he is

currently an Adjunct Professor.His research interests

include the modeling and identi?cation of systems,

electrical machines,and control system

applications.

Hilton Abílio Gründling was born in Santa Maria,

Brazil,in1954.He received the B.Sc.degree from

the Ponti?cal Catholic University of Rio Grande do

Sul,Porto Alegre,Brazil,in1977,the M.Eng.de-

gree from the Federal University of Santa Catarina,

Santa Catarina,Brazil,in1980,and the D.Sc.degree

from the Technological Institute of Aeronautics,S?o

Paulo,Brazil,in1995.

Since1980,he has been with the Federal Univer-

sity of Santa Maria,Rio Grande do Sul,Brazil,where

he is currently a Titular Professor.His research in-terests include robust model reference adaptive control,discrete control,and control system applications.

Vieira-2014-Sensorless Sliding-M

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